1
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING
SPACES OF FINITE GROUPS
RAM'ON J. FLORES
Abstract. In this note we discuss the effect of the BZ=p-nullification P*
*BZ=p
and the BZ=p-cellularization CW BZ=pover classifying spaces of finite gr*
*oups,
and we relate them with the corresponding functors with regard to Moore
spaces, that have been intensively studied in the last years. We descri*
*be
PBZ=pBG by means of a Postnikov fibration, and we classify all finite gr*
*oups G
for which BG is BZ=p-cellular. We also study carefully the analogous fun*
*ctors
in the category of groups, and its relationship with the fundamental gro*
*ups of
PBZ=pBG and CW BZ=pBG.
1.Introduction
Let A be a pointed connected space. P. Bousfield defined in [Bou94 ] the A-
nullification functor PA as the localization Lf with regard to the constant map
f : A -! *. Roughly speaking, if X is another pointed space, PA (X) is the bigg*
*est
üq otientö f X that has no essential map from any suspension of A. A little bit
later, Dror-Farjoun defined ([DF95 ]) the somewhat dual notion of A-cellulariza*
*tion
of a space X as the largest space CW AX endowed of a map CW AX -cw!X such
that every map from a n-suspension of A to X lifts to CW AX via cw. The A-
cellularization of X can also be viewed as the closest approximation of X that *
*can
be built from A taking pointed homotopy colimits. The very close relationship t*
*hat
exists between these two functors was clarified by the work of Chach'olski ([Ch*
*a96 ])
where he explains how to describe anyone of these functors in terms of the othe*
*r,
and he introduces the crucial concept of closed class. Moreover, these functors
have been widely studied and used since they appeared; see for example [Dwy94 ],
[Bou97 ], and [CDI02 ].
Dror-Farjoun also defines the A-homotopy theory of a space, where A and its
suspensions play the role that S0 and its suspensions play in classical homotopy
theory. For example, he defines the A-homotopy groups ßi(X; A) as the homo-
topy classes of pointed maps [ iA, X]*. In this framework, the functors P iA(X)
and CW iA(X) can be viewed, respectively, as the i-th Postnikov section and
the i-connective cover of X. Moreover the spaces for which X ' PA (X) (A-
null spaces) play the role of weakly contractible spaces, and the spaces such t*
*hat
A ' CW AX (A-cellular spaces) are the analogues of the CW-complexes. Indeed,
the A-cellularization is nothing but the A-cellular approximation of X.
Our main interest is BZ=p-homotopy theory. After the positive resolution by
Miller ([Mil84]) of the Sullivan conjecture, there has been a lot of research w*
*hich has
allowed deep knowledge of the space map *(B Z=p, X) when the target is nilpoten*
*t,
and hence it has given clues about the value of the BZ=p-nullification functor *
*over
X; a good survey on this topic can be found in the book [Sch94]. For example,
we have the result of Lannes-Schwartz ([LS89 ]) which points out that if moreov*
*er
the Z=p-cohomology of X has finite type and ß1(X) is finite, then X is BZ=p-null
____________
1Mathematical subject classification: 55P20, 55P80. Key words: (Co)locali*
*zation, finite
groups, Eilenberg-MacLane spaces.
Partially supported by MCYT grant BFM2001-2035.
1
2 RAM'ON J. FLORES
if and only if X is locally finite with regard to the action of the Steenrod al*
*gebra.
Unfortunately, not much is known if we do not impose the nilpotence hypothesis.
We have focused our study in the BZ=p-nullification and BZ=p-cellularization
of classifying spaces of finite groups. Even if the group G is not nilpotent (*
*and
hence BG is not nilpotent as a space, and we cannot apply the mentioned results*
*),
this seems to be an accesible case, because we have a precise description of the
space map *(B Z=p, BG) as the space of group homomorphisms Hom (Z=p, G), which
is homotopically discrete. In particular, this gives a feeling that the value o*
*f these
functors over BG will depend strongly on the group theory of G.
One of the classical invariants that measure the p-primary part of the homot*
*opy
structure of a space X are the homotopy groups with coefficients in Z=p. Recall
that the (n+1)-th homotopy group of X with coefficients in Z=p is defined to be*
* the
group of homotopy classes of pointed maps [M(Z=p, n), X]*, where M(Z=p, n) is t*
*he
corresponding Moore space and n 1. Thus, understand these homotopy groups
amounts to describe the M(Z=p, 1)-homotopy theory of X. This problem has been
attacked succesfully by Bousfield ([Bou97 ]) and Rodr'iguez-Scherer ([RS98 ]), *
*who
describe precisely the M(Z=p, 1)-nullification and M(Z=p, 1)-cellularization of*
* BG.
Then it seems natural to ask if there exists any similar description of the BZ=*
*p-
homotopy theory of BG and if so, what relationship does it have with the mentio*
*ned
p-primary part of the classical homotopy structure of BG.
Our first result gives a partial answer to this question by characterizing t*
*he BZ=p-
nullification of the classifying space of a finite group G by means of a Postni*
*kov
fibration. Recall that Bousfield defines the Z=p-radical T Z=pG of G (sometimes
denoted by OpG in a framework of group theory) as the smallest normal subgroup
of G that contains all the p-torsion.
Theorem 1.1. Let G be a finite group, p a prime number; then, the B Z=p-
nullification PBZ=pBG of BG fits in the fibration sequence:
Y
B (T Z=pG)^q-! PBZ=pBG -! B(G=T Z=pG).
q6=p
where B (T Z=pG)^qdenotes the Bousfield-Kan Z=q-completion of B (T Z=pG) (see
[BK72 ] for the definition and main properties of this functor).
It is easy to see from this result that the only groups G for which BG is BZ*
*=p-
acyclic are the p-groups, and, according to [Lev95], PBZ=pBG has nonzero homoto*
*py
groups in highly arbitrary dimensions if and only if the Z=p-radical of G is a *
*p-group.
Moreover, if G is simple, the BZ=p-nullification of BG is simply-connected, and*
* it
is not hard to prove that if G is nilpotent, PBZ=pBG is an Eilenberg-MacLane sp*
*ace
that is nilpotent too.
If M (Z=p, 1) is a Moore space of dimension 2, the results ([Bou97 ], sectio*
*n 7) and
the previous theorem guarantee that the M (Z=p, 1)-nullification of BG is homot*
*opy
equivalent, via the map induced by the inclusion M (Z=p, 1) ,! BZ=p, to the BZ=*
*p-
nullification of BG. Hence, these functors coincide over classifying spaces of *
*finite
groups, and this proves that the aforementioned relationship between the p-prim*
*ary
part of BG and its BZ=p-homotopy theory is really close. On the other hand, it *
*is
easy to see that these two functors do not coincide over every X: take for exam*
*ple
p = 2 and X = RP2, which is a model for M (Z=2, 1) and is BZ=2-null by Miller's
theorem ([Mil84]).
An important consequence of the previous observation is that a great part of
the results of ([RS98 ]) relative to Moore spaces remain valid for the case of *
*BZ=p-
nullification, and in particular they allow us to obtain a very precise descrip*
*tion of
the value of the acyclic functor ~PBZ=pover BG. We recall that the acyclic func*
*tor
is the colocalization associated to the nullification.
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS3
We finish the study of PBZ=pBG using our description of it for proving (3.7)*
* that
the nullification functors PBZ=p and PBZ=q commute over BG for different primes*
* p
and q (see [RS00 ] for an overview of the problem of commutation in localizatio*
*n the-
ory), and establishing some relations (3.10 and 3.12) between the BZ=p-nullific*
*ation
functor and the Bousfield-Kan Z=p-completion and Z[1=p]-completion.
The second part of this note is devoted to the analysis of the BZ=p-cellular*
*ization
of BG. Our main result on this topic has been the characterization of the class*
* of
finite groups G such that BG is BZ=p-cellular.
Theorem 1.2. Let G be a finite Z=p-cellular group. Then BG is BZ=p-cellular if
and only if G is a p-group generated by order p elements.
The proof relies essentially on a result of Chach'olski (2.4) about preserva*
*tion of
cellularity under fibrations.
In general the BZ=p-cellularization of BG is related very closely with the Z*
*=p-
cellularization CW Z=pG of G in the category of groups (see section 4 for the p*
*recise
definition and [RS98 ] for the original reference) by means of the following le*
*mma:
Lemma 1.3. If G is a finite group, the natural map CW Z=pG -! G induces a
homotopy equivalence CW BZ=pBCW Z=pG ' CW BZ=pBG.
From this result, it is desirable to have an algorithm for computing the Z=p-
cellularization CW Z=pG of G, which allows to restrict ourselves to the calcula*
*tion
of CW BZ=pBG to the case of G Z=p-cellular. According to ([RS98 ], 2.7) the gr*
*oup
CW Z=pG fits in a central extension
0 -! A -! CW Z=pG -! SZ=pG -! 0
where SZ=pG is the the Z=p-socle of G, v.g. the (normal) subgroup of G generated
by the order p elements, and A is the second homotopy groupWof the M (Z=p, 2)-
cellularization of the cofibre Cf of the evaluation map [BZ=p,BG]*BZ=p -! BG.
In fact, it is easily proved that A = ß2(Cf)=T Z=p(ß2(Cf)), so our goal has bee*
*n to
calculate ß2(Cf) and we have obtained the following:
Proposition 1.4. Let G a finite group generated by order p elements, K the kern*
*el
of the evaluation map *Z=p -! G, where the free product is extended to all the
homomorphisms Z=p -! G; then ß2(Cf) = K=[K, *Z=p].
This group is computable, because it is finitely generated and hence we can *
*use
the Reidemeister-Schreier method to obtain a presentation of it. It is worth to
point out that this result is applicable to every finite group G, because of th*
*e fact
that CW Z=pG = CW Z=pSZ=pG and the Z=p-socle is always generated by order p
elements.
Once we have computed A, the universal properties characterizing the previous
extension ([RS98 ], 2.7) allows a complete description of it in most cases, and*
* some-
times give a precise hold of the extension class. On the other hand, we prove t*
*hat
the Z=p-cellularization of G can give a lot of information about some group-the*
*oretic
invariants of the group G, such that its Schur multiplier, or its universal cen*
*tral
extension with coefficients in Z or Z[1=p] (if G is respectively perfect or Z[1*
*=p]-
perfect). In fact, the latter turns out to be the fundamental group of ~PBZ=pB*
*G,
and this is used to prove that the BZ=p acyclic functor öc mmutes" with funda-
mental group, in a similar way that BZ=p-nullification does.
Our study of the cellularization ends up with a description of the fundament*
*al
group of the BZ=p-cellularization of BG as a central extension of G by a p-grou*
*p.
In last section we compute the value of PBZ=p and CW BZ=pover the classifyi*
*ng
spaces of various concrete families of finite groups. In particular, we analyze*
* the
4 RAM'ON J. FLORES
effect of these functors over classifying spaces of dihedral, semidihedral, sym*
*metric
or quaternionic groups.
Notation. In all this note the word "space" will stand for "CW-complex", and
usually we will suppose that these spaces are pointed. The notation X^pwill den*
*ote
Bousfield-Kan Z=p-completion of X, whereas Z[1=p]1 X will stand for the Z[1=p]-
completion.
Acknowledgements. I would like to thank Carlos Broto for all his suggestions*
* and
commentaries and also for his warm encouragement through all the time this work
was done. The results about cellularization of groups were greatly improved in *
*some
interesting discussions with Fernando Muro. I also appreciate some conversations
held with Carles Casacuberta, Emmanuel Dror-Farjoun, Dikran Karagueuzian and
Jer^omeScherer. Finally, I am grateful to the Institute Galil'ee, LAGA, Univers*
*it'e
Paris XIII, for their kind hospitality during the time I spent there.
2.Preliminaries
In this paragraph we recall briefly the definition and properties of the nul*
*lification
functor PA and the cellularization functor CW A. We only comment here the resu*
*lts
that are somewhat relevant for our work; the main references on this topic are
[Bou94 ], [DF95 ] and [Cha96 ].
In all the section, A will stand for a connected space, and all the spaces w*
*ill be
pointed, unless express mention against.
A space X is called A-null if the mapping space map *(A, X) is weakly con-
tractible, which roughly speaking means that every pointed map of a n-suspensio*
*n of
A into X is inessential. The A-nullification PA X (sometimes called A-periodiza*
*tion)
is the only A-null space, up to homotopy equivalence, endowed with a map X -!
PA X which induces, for every A-null space Y , a weak homotopy equivalence
map*(PA X, Y ) ' map *(X, Y ).
In this way it is defined a functor PA : Spaces* -! Spaces* which is coaugmented
and idempotent. We remark that there is no problem for defining the nullificati*
*on
functor in the unpointed category, with no reference to the base point.
There are some constructions of PA , and the easiest is probably the followi*
*ng:
take a space X~ for every ordinal ~; define X0 = X and by induction let X~+1
be the space obtained from X~ gluing cones over all posible homotopy classes of
maps from n-suspensions of A into X~. Then, the A-nullification of X is defined
as the colimit of all these spaces. Other constructions can be found in Bousfi*
*eld
([Bou94 ],2.8) and Chach'olski ([Cha96 ],17.1).
The A-nullification can also be defined as the localization in the sense of *
*Dror-
Farjoun with regard to the trivial map A - ! *, and using that the following
properties are easy to prove ([DF95 ], 1.A.8):
o The natural map PA (X x Y ) -! PA X x PA Y is a homotopy equivalence.
o If X is A-null then it is indeed A-null for every n, and nX is A-null.
o If PA X ' *, then P A X ' *.
o If we have a fibration where the base and the fibre are A-null, then the
total space is A-null.
o If X is n-connected, PA X is n-connected too.
One of the most importants achievements of the work of Dror-Farjoun is the
description of the behaviour of the localization functors with regard to fibrat*
*ions.
In particular, he defines the fibrewise nullification, and proves the following*
* results,
that will be crucial and intensively used in our work:
Theorem 2.1 ([DF95 ],1.H.1 and 3.D.3). Let F -! E -! B be a fibration. Then
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS5
(1)If PA (F ) is contractible, then the induced map PA (E) -! PA (B) is a
homotopy equivalence.
(2)If B is A-null then the fibration is preserved under A-nullification.
The most simple example of nullification functor is the Postnikov n-section,
which is exactly the Sn+1-nullification. Other widely studied examples have been
Quillen plus-construction X -! X+ , which is the nullification with regard to a
large space that is acyclic for a certain homology theory (see [BC99 ], for exa*
*mple),
the nullification with regard to Moore spaces, or the BZ=p-nullification of cla*
*ssifying
spaces of compact Lie groups such that the group of components is a p-group. As
we will see, our work is closely related with these two examples.
To conclude with nullification, we will comment briefly the concept of A-per*
*iodic
equivalence, due to Bousfield:
Definition 2.2. A map f : X -! Y is called an A-periodic equivalence if for any
A-null space Z and any choice basepoints in X, Y , Z, the map f induces a weak
homotopy equivalence map *(Y, Z) ' map *(X, Z). In particular, the coaugmenta-
tion X -! PA X is always an A-periodic equivalence, because the functor PA is
idempotent.
In ([Cha96 ], section 13), one can find the main properties of A-periodic eq*
*uiva-
lences.
In the same way as the A-nullification isolates "the partö f a space that i*
*s not
visible by means of maps nA -! X, the A-cellularization describe to what extent
a space can be built using A as building blocks. We will begin our overhaul of
cellularization defining the concept of cellular space.
Definition 2.3. A space X is called A-cellular if for any choice of basepoint i*
*n X
and for every pointed map f : Y -! Z such that f* : map *(A, Y ) -! map *(A, Z)
is a weak equivalence, we have that the induced map f* : map *(X, Y ) -! map *(*
*X, Z)
is also a weak equivalence. It can be proved that this is equivalent to say tha*
*t X
can be built as an (iterated) pointed homotopy colimit of copies of A.
Hence, the A-cellularization of X is defined as the unique A-cellular space
CW A X (up to homotopy) such that it exists a canonical augmentation cw :
CW A X -! X which induces a weak homotopy equivalence map *(A, CW AX) '
map *(A, X). In particular, the augmentation cw has the following two features
([DF95 ], 2.E.8):
(1)If f : Y -! X induces a weak homotopy equivalence map *(A, Y ) '
map *(A, X), there is a map f0 : CW AX -! Y such that f O f0 is ho-
motopic to cw. Moreover, f0 is unique up to homotopy.
(2)If Z is A-cellular and g : Z -! X is a map, then it exists g0: Z -! CW *
*AX
such that cw O g0 is homotopic to g and g0 is unique up to homotopy.
Dror Farjoun also gave the first two constructions of the functor CW A, the
standard ([DF95 ] 2.E.3), and another one that is more intuitive but it has the
disadvantage of being not functorial ([DF95 ] 2.E.5). However, we will recall h*
*ere
the construction of Chach'olski ([Cha96 ], section 7) because it will be more u*
*seful
for our purposes.
Proposition 2.4. If A is connected, the A-cellularization of X has the homotopy
type of the homotopy fibre of the map j : X - ! LX, where j is the composi-
tionWof the inclusion X ,! Cf into the homotopy cofibre of the evaluation map
[A,X]*A -! X, with the nullification Cf -! P A Cf.
This can be interpreted as a definition of the functor CW A in terms of PA ,
and in fact it is also possible to describe PA in terms of the A-cellularizatio*
*n. It
6 RAM'ON J. FLORES
is worth to point out that any construction of CW A must be worked out in the
pointed category, because it is not possible to define CW A over unpointed spa*
*ces
([Cha96 ], 7.4).
Chach'olski also makes the key observation that A-cellular spaces constitute*
* a
closed class, v.g. a class of spaces that is closed under weak equivalences and*
* pointed
homotopy colimits. In particular, the class of A-cellular spaces is the smalles*
*t closed
class that contains A. Now we list some important properties of A-cellular spac*
*es,
which are nothing but the translations of the corresponding properties of closed
classes; the last one will be particularly important in this note, because it a*
*llows
to build new cellular spaces from old ones. The proofs can be found in ([Cha96 *
*],
section 4):
Proposition 2.5. Let A be a space. Then
o If X is weakly contractible, it is A-cellular.
o If B is A-cellular and X is B-cellular, then X is A-cellular.
o If X is A-cellular, then the n-suspension nA is A-cellular.
o If F -! E -! B is a fibration with a section and F and B are A-cellular,
then E is A-cellular. In particular, the product of two A-cellular space*
*s is
A-cellular, and for every pair of spaces X, Y ; we have a weak equivalen*
*ce
CW A (X x Y ) ' CW AX x CW AY .
In ([DF95 ], 3.5) one can find a lot of examples of interesting A-cellular s*
*paces.
For example, the James construction A is A-cellular, the Dold-Thom functor
SP 1A is A-cellular, the classifying space of a group BG is G-cellular for eve*
*ry
group G, etc. In particular, he proves that for every n 1 the BZ=p-cellulariz*
*ation
of BZ=pn is BZ=p, a fact that can be considered a starting point for our work.
It is also worth to recall the concept of A-cellular equivalence:
Definition 2.6. A map X -! Y is called an A-cellular equivalence if it induces a
weak equivalence map *(A, X) ' map *(A, Y ).
The main properties of the A-cellular equivalences can be found in ([Cha96 ],
section 6).
We finish this sample, which certainly doesn't pretend to be exhaustive, by
commenting on the relationship between the functors PA and CW A. For this is
necessary to define the acyclics, which at any rate have their own interest.
Definition 2.7. An space X is called A-acyclic if PA X is contractible. The fun*
*ctor
P~A : Spaces* -! Spaces* which sends every space to the homotopy fiber of its
A-nullification is augmented and idempotent (in fact, it is a colocalization) a*
*nd its
image is the class of A-acyclic spaces.
The class of A-acyclic spaces was the crucial ingredient that Chach'olski us*
*ed for
stating the main result of [Cha96 ], which strongly generalized a theorem of Dr*
*or-
Farjoun, and gave an amazingly sharp description of to what extent the functors
PA and CW A can be considered üd al":
Theorem 2.8. Let A be a space. Then
(1)A space X is A-null if and only if its A-cellularization is a point.
(2)A space X is A-acyclic if and only if it belongs to the smallest closed *
*class
that contains A and is closed by extension by fibrations. In particular,*
* every
A-cellular space is A-acyclic.
We recall that a closed class C is closed by extensions by fibrations if for*
* every
fibration F -! E -! B such that F 2 C and B 2 C, we have E 2 C.
This result and ([DF95 ] 9.A.6), which leave the taste that the A-cellulariz*
*ation
is a kind of mixing process between the functors ~PA and ~P A, have made much
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS7
more accesible the computation of the value of the A-cellularization of a space*
*, and
have greatly stimulated the research on this field. Among the most recent works
on it, we can quote [CDI02 ], where the authors generalize the notion of dimens*
*ion
of a CW-complex to the A-cellular framework, or [DGI02 ], where they generalize
the notion of cellularization to algebraic categories of R-modules.
3.B Z=p-nullification of classifying spaces of finite groups
3.1. Computing the nullification of BG. Let G be a finite group, p a prime
number. As we have stated in the introduction, our interest has been focused on
studying the p-primary part of the classifying space of G using the functors PB*
*Z=p
and CW BZ=p; in this section we will be concerned with the former one. So, our
first result on this topic is a characterization of the space PBZ=pBG by means *
*of a
Postnikov fibration. As far as we know, the unique description already done of *
*this
space is for the case of G nilpotent, and in this case the proof is really easy*
*: if H
is the p-torsion subgroup of G, it is enough to BZ=p-nullify the fibration
BH -! BG -! B(G=H)
for obtaining that PBZ=pBG ' B(G=H). The difficulty of the general case take ro*
*ot
in the fact that in general the minimal subgroup that contains the p-torsion ha*
*ve
elements that are not of p-torsion. We get round the trouble identifying first *
*the
case in which the BZ=p-nullification is simply-connected, and then passing to t*
*he
general case.
Proposition 3.1. Let G be a finite group, p prime. Suppose thatQG has no non-
trivial quotients of order prime to p. Then we have PBZ=pBG ' q6=pBG^q(which
is in fact homotopy equivalent to Z[1=p]1 BG).
Q
Proof. First of all, we have to prove that q6=pBG^qis a BZ=p-null space. But *
*this
is clear because
map*(B Z=p, BG^q) ' map *((B Z=p)^q, BG^q) ' *
where the first equivalence holds because BG^qis Z=q-complete.
So, now we must see that if X is another BZ=p-null space, it exists a homoto*
*py
equivalence Y
map *(B G, X) ' map *( B G^q, X).
q=2S
We will consider two cases.
First, we will suppose that X is a simply-connected space. In this case, Sul*
*livan's
arithmetic square gives us a homotopy equivalence
Y
map *(B G, X) ' map *(B G, X^q).
q prime
We shall check that map *(B G, X^p) is contractible. The space X^pis Z=p-comple*
*te,
so we have an equivalence map *(B G, X^p) ' map *(B G^p, X^p). Using Jackowski-
McClure-Oliver subgroup decomposition ([JMO92 ], see also [Dwy97 ]), we obtain
that
map *(B G^p, X^p) ' map *((hocolimOCfiC)^p, X^p)
where fiC is a functor whose values have the homotopy type of classifying spaces
of p-subgroups of G, and OC is a Z=p-acyclic category. Again, because of X^pis
Z=p-complete, we have
map*((hocolimOCfiC)^p, X^p) ' map *(hocolimOCfiC, X^p).
8 RAM'ON J. FLORES
By the classical result of Bousfield-Kan ([BK72 ], XII, 4.1), we have
map *(hocolimOCfiC, X^p) ' holimOopCmap*(fiC(-), X^p).
But X is BZ=p-null, and by a theorem of Miller ([Mil84] 9.9), its Z=p-completion
is; so, as the functor fiC take its image over p-groups, the space map *(fiC(-)*
*, X^p)
is contractible for every value of the mentioned functor. This means that
map *(B G, X^p) ' holimOCop* ' *,
as we wanted. So, we have now the following string of weak equivalences:
Y Y (*)
map *(B G, X) ' map *(B G, X^q) ' map *(B G, X^q) '
q6=p q6=p
Y (**) Y
map *(B G^q, X^q) ' map *( B G^q, X)
q6=p q6=p
where the equivalence (*) holds because of X is simply-connected (and therefore*
* X^q
is Z=q-complete for every q) and (**) holds because of the space map *(B G^q, X*
*^r)
is contractible if q and r are different primes.
Now we can attack the general case. Let X be a BZ=p-null space. If X~ is the
universal cover of X, we have the Postnikov fibration
X~- ! X -h! Bß1(X).
Our first goal will be to see that the map
map *(B G, X) -h"!map*(B G, Bß1(X))
takes every map of the space map *(B G, X) to the homotopy class of the constant
map. So, let f :BG -! X be such a map, and consider the following commutative
diagram
vX::
vvv
fvvv |h|
vvv fflffl|
BG __h"f//_Bß1(X)
We must prove that h"f ' *, so let g : BZ=p -! BG be a continuous map. As X is a
B Z=p-null space, we know the composition f O g ' *, and in particular h"f O g *
*' *.
On the other hand, there exist two group homomorphisms, ~ : Z=p -! G and
æ : G -! ß1(X) such that B ~ ' g and B æ ' h"f. Thus, it is clear that the
composition
Z=p -~! G -æ! ß1(X)
is the zero homomorphism, and this happen for every homomorphism Z=p -! G;
so, we obtain Im æ should be a quotient of G whose order is coprime to p. By our
hypothesis, G does not have such nontrivial quotients, so æ is zero, and its in*
*duced
map at the level of classifying spaces is homotopic to the constant, as we want*
*ed to
know. Consider, then, the following diagram, where the left column is a fibrati*
*on,
the horizontal maps are all induced by the product of the Z=q-completions of BG,
and ~Xis the universal cover of X:
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS9
Q '
map *( q6=pBG^q, ~X)________//_map*(B G, ~X)
|(2)| |(3)|
Q fflffl| (1) fflffl|
map *( q6=pBG^q, X)_________//_map*(B G, X)
|h"| |h"|
Q fflffl| fflffl|
map*( q6=pBG^q, Bß1(X))____//map*(B G, Bß1(X))c
The top-horizontal map is an equivalence by the first case done before and
([ABN94 ] 9.7), and the down-right map has been seen to take values in the com-
ponent of the constant map, so it is a fibration. ItQis known that this compo-
nentQis contractible, and the same is true for map *( q6=pBG^q, Bß1(X)), becau*
*se
q6=pBG^qis 1-connected and Bß1(X) is an Eilenberg-MacLane space. Hence, the
maps (2) and (3) are weak equivalences; by the commutativity of the diagram, th*
*is
means that (1) is a weak equivalence, and we have finished.
For the general case of the theorem we will need to identify in some way the
p-torsion of G, and this will lead us to the concept of Z=p-radical.
Definition 3.2. Let G be a finite group, p a prime. The Z=p-radical of G (some-
times called the Z=p-isolator) is the minimal normal subgroup TZ=pG that contai*
*ns
all the p-torsion elements of G.
The following features of this subgroup are easy to prove:
o The index of TZ=pG in G is coprime with p, and TZ=pG is minimal among
the normal subgroups of G for which this condition holds.
o T Z=pG is a characteristic subgroup of G, v.g., every automorphism of G
reduces by restriction to an automorphism of TZ=pG.
o T Z=pG has no normal subgroups whose index in TZ=pG is coprime with p.
Now we are prepared to prove the general case of the theorem:
Theorem 3.3. Let G be a finite group, p a prime; then we have that the BZ=p-
nullification of BG fits in the following fibration sequence:
Y
B (T Z=pG)^q-! PBZ=pBG -! B(G=T Z=pG).
q6=p
Proof. The Z=p-radical is normal in G, so we can consider the fibration of clas*
*sifying
spaces
BT Z=pG -! BG -! BT Z=pG.
The quotient group G=T Z=pG has order coprime with p, so its classifying space *
*is
Z=p-null. Now, by (2.1), the sequence of nullifications
PZ=pBT Z=pG -! PBZ=pBG -! B(G=T Z=pG)
Q
is a fibration sequence. But PZ=pBT Z=pG ' q6=pB(T Z=pG)^qby 3.1, so the theo-
rem is proved.
Recall that if G is a finite group and p is a prime number, it is known ([BK*
*72 ],
II.5, see also [Lev95]) that the fundamental group of BG^qis the quotient of G *
*by its
p-perfect maximal normal subgroup Op(G); in particular, the fundamental group
10 RAM'ON J. FLORES
Q
is a p-group. So, in the case of the theorem, this means that q6=pB(T Z=pG)^q*
*is a
simply-connected space, and thus it is the universal cover of PBZ=pBG.
If S = p1. .p.ris a finite collection of prime numbers, it can be defined the
S-radical of G in the same lines of 3.2 as the minimal normal subgroup TSG of G
that contains all the S-torsion. This group verifies analogous properties for t*
*hose
previously quoted for TZ=pG, and in fact it is a normal subgroup of the Z=pi-ra*
*dical
for every pi2 S. Using this object, we can establish the following generalizati*
*on of
the previous theorem, that can be proved using exactly the same line of reason *
*as
before:
Proposition 3.4. Let G be a finite group, S = p1. .p.na finite collection of pr*
*ime
numbers, and W = BZ=p1 _ . ._.BZ=pn; then we have that the W -nullification of
B G fits in the following fibration sequence:
Y
B(T SG)^q-! PW BG -! B(G=T SG).
q=2S
Following ([RS00 ] 1.1), it is enough to consider only the case in which the*
* primes
are different, because there is a homotopy equivalence PBZ=pBG ' PBZ=p_BZ=pBG.
It is also interesting to note that the BZ=p-nullification of the classifyin*
*g space
of a finite simple group is nothing but a completion:
Corollary 3.5. If p is a prime number and G is a finite simple group, then we
have PBZ=pBG = Z[1=p]1 BG.
Proof. It is a direct consequence of the fibration that gives the theorem 3.3.
It is greatly remarkable the fact that, if M(Z=p, 1) is a 2-dimensional Moore
space, then the inclusion M(Z=p, 1) ,! B Z=p induces a map PM(Z=p,1)BG -!
PM(BZ=p)BG which according to 3.3 and ([RS98 ], 1.3) is a homotopy equivalence.
In other words, this statement tells us that the BZ=p-nullification of BG depen*
*ds
only on the 2-skeleton of BG. On the other hand, it proves the following beauti*
*ful
result, that concerns localization of groups:
Corollary 3.6. If G is a finite group and p is a prime number, LZ=pß1(B G) is
isomorphic to ß1(PBZ=pBG). Here LZ=p denotes the usual localization of G with
regard to the null map Z=p -! *.
Proof. It only must be pointed out that ß1(PM(BZ=p)BG) ' G=T Z=pG ' LZ=pG.
See ([Cas94], 3.2) for details.
In section 4 we will use these last results for giving a precise description*
* of the
fundamental group of ~PBZ=pBG, a way of compute it and a characterization of the
finite groups whose classifying space is BZ=p-acyclic.
We would like to finish this section by mentioning the article [Dwy94 ], whe*
*re
the author proves that the BZ=p-nullification of the classifying space of a com*
*pact
Lie group G whose group of components is a p-group is homotopy equivalent to its
Z[1=p]-localization. We consider our work complementary to that, and it would be
desirable to find a way to arrange all these data to find a description of PBZ=*
*pBG
for every compact Lie group G.
3.2. Commutation of the nullification functors. It is known that localization
functors usually do not commute, not even in the case of nullifications. There *
*are
several examples of this in [RS00 ], where the authors also try to elucidate wh*
*at
happens if we apply in succession two localization functors to a certain space.
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS11
We will prove now that for different primes p and q, the functors PBZ=p and
PBZ=q do commute if we apply them over BG. In view of what we have said, this
is an interesting exception to the general case.
Proposition 3.7. Let G be a finite group, p and q different primes. Then we have
homotopy equivalences
PBZ=pPBZ=qBG ' PBZ=qPBZ=pBG ' PBZ=p_BZ=qBG.
Proof. The proof of this result will be divided in two cases: in the first one *
*we will
suppose that G coincides with its S-radical for S = {p, q}, and then we will pa*
*ss
to the general case.
So, let G = TSG. We will prove that PBZ=pPBZ=qBG is homotopy equivalent to
PBZ=p_BZ=qBG and the other equivalence will follow interchanging the roles of p*
* and
q. We want to check that PBZ=p_BZ=qBG is the BZ=p-nullification of PBZ=qBG, i.e.
PBZ=p_BZ=qBG is BZ=p-null, and for every BZ=p-null space X we have a homotopy
equivalence
map *(PBZ=p_BZ=qBG, X) ' map *(PBZ=qBG, X)
which should be given by a coaugmentation PBZ=qBG -! PBZ=p_BZ=qBG. The
first statement is trivial, because PBZ=p_BZ=qBG is by definition a BZ=p-null s*
*pace.
So we only must verify the previously mentioned homotopy equivalence between
the mapping spaces. So, let X be a BZ=p-null space.
First, as PBZ=p_BZ=qBG is BZ=q-null, we have a natural map
PBZ=qBG -! PBZ=p_BZ=qBG
which induces another one between the mapping spaces
map*(PBZ=p_BZ=qBG, Y ) -! map *(PBZ=qBG, Y )
for every space Y . So, if we consider the BZ=p-null space X and its universal *
*cover
X~ we have the following commutative diagram:
(3.2.1) map *(PBZ=p_BZ=qBG, ~X)___________//map*(PBZ=qBG, ~X)
| |
| |
fflffl| fflffl|
map *(PBZ=p_BZ=qBG, X)____________//_map*(PBZ=qBG, Y )
p~p,q|| |~pq|
fflffl| r" fflffl|
map *(PBZ=p_BZ=qBG, Bß1(X))~pp,q___//map*(PBZ=qBG, Bß1(X))~pq
For proving that the columns are fibrations, we must check that the maps ~pp*
*,qy
p~qtake value in both cases in the component of the constant map. The first one*
* is
trivial, because by 2.1, the space PBZ=p_BZ=qBG is simply connected. In the oth*
*er
case, we must verify that if we have a map f : PBZ=qBG -! X, the composition
with the projection ß : X -! Bß1(X) is homotopic to the constant map. We will
denote by g the composition
B G -! PBZ=qBG -f! X -ß! Bß1(X),
where B G -! PBZ=qBG is the coaugmentation. It is clear that for every map
B Z=p -! BG the composition
B Z=p -! BG -g! Bß1(X)
must be inessential, because X is B Z=p-null, and same happens for every map
B Z=q -! BG, in this case because PBZ=qBG is BZ=q-null. As G has no quotients
12 RAM'ON J. FLORES
coprimes with pq, the map Bg is trivial. Moreover, g factors through the projec*
*tion
G -! G=T Z=qG, and the induced map G=T Z=qG -! ß1(X) is trivial too. Hence,
the commutative diagram
PBZ=qBG _f____//_X___ß//_Bß1(X)55
jjj
| *jjjjj
| jjjj
fflffl|jjj
B (G=T Z=qG)
where PBZ=qBG -! B (G=T Z=qG) is the natural projection over B ß1(PBZ=qBG)
proves that the composition ß O f is trivial, as we wanted.
To finish the proof of the first case, we must see the isomorphism of the top
horizontal level of the diagram 3.2.1. We know ([ABN94 ], 9.4) that if X is a *
*BZ=p-
null space, its universal cover is too, so we only need to check that the previ*
*ously
described map
map *(PBZ=p_BZ=qBG, X) -! map *(PBZ=qBG, X)
is a homotopy equivalence if X is a simply-connected, BZ=p-null space. As X is
simply-connected, it is homotopy equivalent to the pullback of the Sullivan ari*
*th-
metic square ([BK72 ] V,6). By 3.3 and 3.4, the rationalizations of PBZ=p_BZ=qBG
and PBZ=qBG are homotopy equivalent to a point, so
Y
map *(PBZ=p_BZ=qBG, X) ' map *(PBZ=p_BZ=qBG, X^r),
r prime
and same is true for maps which come from PBZ=qBG.
It is clear that
Y Y
map *(PBZ=qBG, X^r) ' map *(PBZ=qBG, X^r)
r prime r prime
and,Qas X^ris Z=r-complete for every prime r, the latter is homotopy equivalent
to r primemap*((PBZ=qBG)^r, X^r). If r 6= q, by the result (3.10) we have that
(PBZ=qBG)^r' BG^r, and on the other hand, the Z=q-completion of the Postnikov
fibration of (3.3) gives us that (PBZ=qBG)^qis contractible (see 3.12). In addi*
*tion,
observe that map *(B G^p, X^p) is contractible too, because X is BZ=p-null (see*
* the
proof of 3.3). So we obtain
Y
map*(PBZ=qBG, X) ' map *(B G^r, X^r).
r6=p,q
In the other case, we have again
Y
map *(PBZ=p_BZ=qBG, X) ' map *((PBZ=p_BZ=qBG)^r, X^r).
r prime
Just like before, if r 6= p, q, we have a homotopy equivalence (PBZ=p_BZ=qBG)^r'
B G^r, and on the other hand, using again the Postnikov fibration of (3.3),
(PBZ=p_BZ=qBG)^p' * ' (PBZ=p_BZ=qBG)^q.
So we have
Y
map*(PBZ=p_BZ=qBG, X) ' map *(B G^r, X^r) ' map *(PBZ=qBG, X)
r6=p,q
and this chain of equivalences finishes the proof of the first case.
Let G now be a finite group, and let us consider the fibration
BT SG -! BG -! B(G=T SG).
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS13
As the base is B Z=p _ BZ=q-null, the fibration is preserved after B Z=p _ BZ=q-
nullification (2.1), and also after BZ=p-nullification and BZ=q-nullification. *
*More-
over, PBZ=p_BZ=qBG is BZ=p-null and BZ=q-null, and so we have a commutative
diagram where the rows are fibrations:
PBZ=p_BZ=qBTSOGO_____//_PBZ=p_BZ=qBG____//_B(G=TOSG)OOO
| | |
| | |Id
| | |
P BZ=pPBZ=qBT SG _____//PBZ=pPBZ=qBG_______//BTSG.
Now, the left vertical map is an equivalence by the case already proved, so *
*the
natural map PBZ=pPBZ=qBG -! PBZ=p_BZ=qBG is an equivalence too and we have
finished.
We finish the section by showing a slight generalization of the previous pro*
*posi-
tion.
Corollary 3.8. Let p1. .p.rand q1. .q.sbe two families of prime numbers, and
denote W = BZ=p1 _ . ._.BZ=pr and W 0= BZ=q1 _ . ._.BZ=qs; then we have
PW P0WBG ' PW_W0 B G ' P0WPW BG.
Proof. It is proved exactly in the lines of the previous theorem. We leave the *
*details
to the reader.
3.3. Relation between the nullification and the completion. It seems quite
natural to ask for the relation between the effect on classifying spaces of fin*
*ite
groups of the B Z=p-nullification and the Z[1=p]1 -completion, because these two
functors "kill" the p-primary part of BG. We have already seen, for instance, t*
*hat
they coincide if G is a simple group, but now we will see by means of an easy
example that this is not always true.
Example 3.9. Consider the dihedral group D15, which is isomorphic to the semidi-
rect product Z=15 o Z=2. It will be seen in 5.1 that the BZ=3-nullification of *
*BD15
is homotopy equivalent to BD10.
On the other hand, the group Z=15 is normal in the semidirect product, and we
can consider the associated fibration
BZ=15 -! BD15- ! BZ=2.
The space BZ=2 is Z=2-complete, so this fibration is preserved by Z=2-completio*
*n,
and we obtain the homotopy equivalence (B D15)^2' BZ=2.
Now, it is obvious that ß1(PBZ=3BD15) = D10, while ß1((B D15)^5x BZ=2) =
Z=2, because (B D15)^5is simply-connected (D15 has no quotient groups that are
5-groups). So, in this case, we obtain that the BZ=3-nullification cannot be t*
*he
product of the Z=2-completion and the Z=5-completion of BD15, and in particular
PBZ=3BD156= Z[1=3]1 BD15, as we wanted to see.
Now we will study what happens if we apply in succession the BZ=p-nullificat*
*ion
and Z=q-completion functors to BG, in the two possible orders, and for primes p
and q not necessarily different. We come to the conclusion that the functor PBZ*
*=p
is quite sensitive, in the sense that it kills the p-primary structure of BG le*
*aving
untouched the q-primary part which is detected by the Z=q-completion functor.
We consider first the case p 6= q.
14 RAM'ON J. FLORES
Proposition 3.10. Let G be a finite group, p and q different primes. Then we
have homotopy equivalences
PBZ=p(B G^q) ' BG^q' (PBZ=pBG)^q
Proof. Every map B Z=p -! B G^qfactors through the Z=q-completion of B Z=p,
because BG^qis Z=q-complete. But (B Z=p)^qis contractible, so BG^qis BZ=p-null
and the first equivalence is proved.
For the second, notice that the map BZ=p -! * is a Z=q-equivalence, and so,
the canonical coaugmentation BG -! PBZ=pBG is again a Z=q-equivalence.
Remark 3.11. The arguments of the last proof remain valid if we replace BG by
any Z=q-good space.
If we now consider the case p = q, we obtain the following:
Proposition 3.12. If G is a finite group and p is a prime number, then PBZ=p(B *
*G^p)
is contractible, and the same happens to (PBZ=pBG)^p.
Proof. We must check that, for every BZ=p-null space, the space map *(B G^p, X)*
* is
contractible. Consider the Postnikov fibration
X~- h!X -f! Bß1(X).
Recall that if X is BZ=p-null, its universal cover ~Xis BZ=p-null too (see [*
*ABN94 ],
9.7). Now we consider a finite group G such that BG^pis a simply-connected spac*
*e.
In this case we have map *(B G^p, Bß1(X)) ' *, so for proving that PBZ=pBG^pis
contractible too we only must check map *(B G^p, ~X) is. But this is proved exa*
*ctly
in the same way as in the proof of (3.1), taking care of the fact that if q 6= *
*p,
map *(B G^p, ~X^q) is contractible.
Now, let G be any finite group. If we denote by Op(G) the p-perfect maximal
normal subgroup of G, it is known that ß1(B G^p) is isomorphic to G=Op(G). Now
we have a fibration sequence:
BOp(G) -! BG -! B(G=Op(G)).
The quotient G=Op(G) is a p-group, so B(G=Op(G)) is Z=p-complete, and we
have the correspondent sequence of the Z=p-completions is a fibration:
BOp(G)^p-! BG^p-! B(G=Op(G))
As Op(G) has no quotients of order a power of p, the Z=p-completion BOp(G)^p
is a simply-connected space, and so the last fibration is the Postnikov fibrati*
*on
of BG^p. But we have proven that the fiber BOp(G)^pis BZ=p-acyclic, so by 2.1,
PBZ=p(B G^p) is homotopy equivalent to PBZ=p(B (G=Op(G))), and this last one is
contractible because G=Op(G) is a p-group. So we have finished the proof of the
first statement.
Q For proving the second, notice that the universal cover of PBZ=pBG, that is
q6=p(B G)^qfor q prime, is Z=p-homology equivalent to a point. So by the fib*
*re
lemma 5.1 of [BK72 ], the Postnikov fibration
Y
(B G)^q-! PBZ=pBG -! B(G=T Z=pG)
q6=p
is preserved by Z=p-completion. But the Z=p-completions of the base space and
the fibre are contractible, so (PBZ=pBG)^pis contractible too and we have finis*
*hed.
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS15
In conclusion, we will establish the relationship between the BZ=p-nullifica*
*tion
and the Bousfield-Kan completion with coefficients in the ring Z[1=p].
Proposition 3.13. Let G be a finite group, p and q two different primes. Then
the following relations hold:
(1)Z[1=p]1 PBZ=pBG ' PBZ=pZ[1=p]1 BG ' Z[1=p]1 BG.
(2)Z[1=p]1 PBZ=qBG ' PBZ=qZ[1=p]1 BG ' Z[1=p, 1=q]1 BG
Proof. It is an immediate consequence of the previous results of this section, *
*taking
into account the results ([BK72 ], VII, 4.2 and 4.3) that allow us to express t*
*he
Z[1=p]-completion of PBZ=pBG as the product of their Z=p0-completions in the re*
*st
of primes.
4. Cellularization
Let G be a finite group. In the proposition 3.6 we have seen that the BZ=p-
nullification of BG is intimately related with the Z=p-localization of G as a g*
*roup.
In this way, it turns out to be interesting to study the Z=p-cellularization of*
* the
group G for obtaining information about the BZ=p-cellularization of BG. This is
with broad strokes our approach to this subject, and it is worth to recall the *
*main
definitions concerning the cellularization in the category of groups. Recall th*
*at a
group G is Z=p-cellular if and only if it can be built from G by (maybe iterate*
*d)
colimits, and the Z=p-cellularization of G is the unique Z=p-cellular group CW *
*Z=pG
endowed with an augmentation CW Z=pG -! G which induces an isomorphism
Hom (Z=p, CW Z=pG) ' Hom (Z=p, G). This concept was first defined in [RS98 ] and
mainly used for describing the cellularization with regard to Moore spaces.
We begin our study by showing that the problem of the B Z=p-cellularization
of classifying spaces of finite groups can be reduced to the problem of the BZ=*
*p-
cellularization of classifying spaces of finite Z=p-cellular groups.
Proposition 4.1. If G is a finite group, the natural map CW Z=pG -! G induces
a homotopy equivalence CW BZ=pBCW Z=pG ' CW BZ=pBG.
Proof. By the functoriality of CW BZ=p, the mentioned map CW Z=pG -! G in-
duces another one CW BZ=pBCW Z=pG -! CW BZ=pBG which composed with the
canonical augmentation gives rise to the map f : CW BZ=pBCW Z=pG -! BG. The
source is BZ=p-cellular, so we only need to see that the last map induces a weak
equivalence between the pointed mapping spaces map *(B Z=p, CW BZ=pBCW Z=pG)
and map *(B Z=p, BG). This is proved by the following string of weak equivalenc*
*es:
map *(B Z=p, CW BZ=pBCW Z=pG) ' map *(B Z=p, BCW Z=pG) '
Hom (Z=p, CW Z=pG) ' Hom (Z=p, G) ' map *(B Z=p, BG).
what finishes the proof. Note that we have build explicitly the natural augment*
*ation
f.
Hence, it becomes interesting to find appropriate tools for computing the Z=*
*p-
cellularization of a finite group G. Aiming to this, we need to recall the foll*
*owing
concept of group theory, that will be crucial in the sequel:
Definition 4.2. If G is a finite group, the Z=p-socle SZ=pG of G is the subgroup
of G generated by the order p elements of G.
It can be seen that this subgroup is always normal and characteristic, and i*
*t is
contained in the Z=p-radical TZ=pG. In fact, the following holds:
Proposition 4.3. If G is a Z=p-cellular group, then G = SZ=pG = TZ=pG.
16 RAM'ON J. FLORES
Proof. If G is Z=p-cellular, it is a colimit of Z=p0s, and hence it is generate*
*d by
order p elements. As the Z=p-socle is the group generated by the order p elemen*
*ts
of G, G = SZ=pG, and the other equality is obvious from the inclusions SZ=pG
T Z=pG G.
For our work, the most relevant property of the Z=p-socle is the fact that
the inclusion SZ=pG G always induces an isomorphism Hom (Z=p, SZ=pG) '
Hom (Z=p, G), and this implies that CW Z=pG ' CW Z=pSZ=pG. This last asser-
tion shows that the computation of the Z=p-cellularization of groups can be aga*
*in
reduced to the case of groups generated by order p elements. Moreover, it is wo*
*rth
to point out that we are interested only in finite groups, and in this case it *
*is not
hard to calculate the Z=p-socle of a group starting from a presentation of G (u*
*sing
GAP, for example).
Now, the main tool we are going to use in our description of the Z=p-cellula*
*rization
is the following version of a theorem of Rodr'iguez-Scherer ([RS98 ] 2.7):
Theorem 4.4. For each group G, there is a central extension
0 -! A -! CW Z=pG -! SZ=pG -! 0
such that A has no order p elements and is universal with regard to this proper*
*ty.
The key case is G finite and Z=p-cellular; thus, G = SZ=pG and the essen-
tial problem here is computing A. According to (2.4), this group is the second
homotopy groupWof the M(Z=p, 1)-nullification of the cofibre Cf of the evalua-
tion map f : [M(Z=p,1),BG]M(Z=p, 1) -! BG, where M(Z=p, 1) stands here for
a two-dimensional Moore space. It is not hard to see that ß2(P M(Z=p,1)Cf) =
ß2(Cf)=T Z=pß2(Cf), so our problem is to describe the second homotopy group
of this homotopy cofibre. As G is generated by order p elements, Cf is simply-
connected, and then ß2(Cf) = H2(Cf), the Schur multilplier of the cofibre. Using
this property, we have computed the group in the following way:
Proposition 4.5. Let G be a finite group generated by order p elements, denote *
*by
H the free product *Z=p extended over all the homomorphisms Z -! G, and let K
be the kernel of the evaluation map H -! G. If Cf is the cofiber of the evaluat*
*ion
map at the level of classifying spaces, we have ß2(Cf) = K=[K, H].
Proof. It is an easy consequence of ([BL87 ], 3.4) taking, in the notation of t*
*here,
P = H,M = H and N = K.
As we said in the introduction, using this proposition it is not hard to cal*
*cu-
late a presentation of the group ß2(Cf) from presentations of K and H using the
Reidemeister-Schreier method ([MKS76 ], 2.3). In this way we obtained the value
of ß2(Cf) in the case G = P SL(2, 3) the tetrahedral group (see 5.2) and the co*
*m-
putation of this pathological example served as motivation for further work. Th*
*is
formula is particularly useful for calculating the Z=p-cellularization of G if *
*we don't
know the Schur multiplier of the group G, and indeed gives a way of computing
directly this last invariant:
Corollary 4.6. With the notation of the previous proposition, H2(G) = K \
[H, H]=[K, H].
Proof. It is an immediate consequence of the previous proposition and the Mayer-
Vietoris sequence associated to the cofibration f.
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS17
On the other hand, if the second homology group of G is known, the computa-
tions are easily simplified using the following lemma, which in fact will lead *
*us to
a complete classification of Z=p-cellular groups:
Lemma 4.7. Let G be a finite group generated by order p elements. Then the
group A of 4.4 is isomorphic to H2(G)=T Z=p(H2(G)).
Proof. The Mayer-Vietoris sequence of the cofibration f has the form
M
0 -! H2(G) -! ß2(Cf) -! Z=p -! 0.
But as A = ß2(Cf)=T Z=pß2(Cf), the result follows.
Proposition 4.8. Let G be a finite group. Then G is Z=p-cellular if and only if*
* is
generated by order p elements and H2(G) is a p-group.
Proof. If G is Z=p-cellular, it a colimit of Z=p0s, and hence is generated by o*
*rder
p elements. According to 4.4 and the previous proposition, H2G must be equal to
its Z=p-radical, and hence it must be a p-group. Reciprocally, if G is generate*
*d by
order p elements, then G = SZ=pG, and the fact that H2(G) is a p-group implies
that the group A of 4.4 is trivial; so G is Z=p-cellular.
Once we have computed A, the only thing that remains is to know what is the
extension that identifies the Z=p-cellularization. This is usually not hard, be*
*cause
the key result 4.4 describes that extension with great precision. Furthermore,*
* in
some favorable cases, we can walk one step up and find explicitly the cohomology
class associated to this extension. For this reason we turn now our attention *
*to
perfect groups. Recall that a group G is called perfect if it is equal to its c*
*ommutator
subgroup or, equivalently, if the first integer homology group is trivial.
Proposition 4.9. Let G be a finite perfect group generated by order p elements,
let
0 -! H2G -! ~G-! G -! 0
be the universal central extension of G, let B be the quotient of H2G by the p-*
*torsion
(which is in fact a subgroup of H2G), and let
0 -! B -! H -! G -! 0
be the central extension E induced by the previous one. Then, the latter is equ*
*ivalent
to the extension of the theorem 4.4, and in particular H is isomorphic to the Z*
*=p-
cellularization of G.
Proof. The group B has no p-torsion, so by the universality of the extension E0*
*of
4.4 that defines the Z=p-cellularization there is an unique morphism of extensi*
*ons
E0 -! E that is the identity over G.
On the other hand, it is clear that for any central extension whose kernel h*
*as
not p-torsion, the unique morphism that come from the universal central extensi*
*on
to this one factors through E. This proves that there is again an unique morphi*
*sm
over G from E to E0, and is easy to check, by universality, that the two morphi*
*sms
that we have defined are one inverse to each other. Hence, the extensions E and*
* E0
are equivalent and we have finished.
Hence, if G is a perfect group, the cohomology class in H2(G; A) that corres*
*pond
to the extension E0 that defines the Z=p-cellularization of G is the image of t*
*he
identity map of A under the universal coefficient isomorphism
Hom (A, A) ' Hom (H2G, A) ' H2(G; A).
18 RAM'ON J. FLORES
On the other hand, the methods of computation of CW Z=pG developed above
can be reinterpreted in a framework of group theory as tools for describing the
universal central extension of a finite perfect group G. For example, we can g*
*et
this easy and interesting consequence:
Corollary 4.10. If G is a perfect group generated by order p elements and whose
Schur multiplier has no p-torsion, then the Z=p-cellularization of G is isomorp*
*hic
to its universal covering group ~G.
A somewhat similar line of reasoning can be applied sometimes to non-perfect
groups, as we see in the next proposition:
Proposition 4.11. Let p be an odd prime, G a finite group such that the Schur
multiplier of its Z=p-socle is Z=2. Then the central extension of 4.4 that iden*
*tify
the Z=p-cellularization of G is the one that is not trivial.
Proof. The Z=p-socle SZ=pG of G is generated by order p elements, so its abelia*
*n-
ization is an elementary abelian p-group. Hence, we have H1(SabZ=pG, Z=2) = 0. *
*By
the universal coefficient theorem, we have the isomorphisms
H2(SZ=pG, Z=2) ' Hom (H2(SZ=pG), H2(SZ=pG)) = Hom (Z=2, Z=2) ' Z=2.
This implies that there are only two central extensions of SZ=pG by Z=2. Now
we observe that the group defined by the trivial central extension Z=2 x SZ=pG *
*is
not Z=p-cellular, because it cannot be generated by order p elements. Thus, the
group defined by the nontrivial extension that corresponds to the identity elem*
*ent
in Hom (H2(SZ=pG), H2(SZ=pG)) is the Z=p-cellularization of G.
This result will be very useful in certain cases, as we will see in the next*
* section.
Once we have studied carefully the Z=p-cellularization functor in the category *
*of
finite groups, it turns out to be interesting to relate it to the other group c*
*olocal-
ization involved in this work, namely the homotopy fiber ~LZ=pG of the localiza*
*tion
map G -! LZ=pG described in 3.1. This is established in the next proposition.
Proposition 4.12. If G is a finite group such that T Z=pG = SZ=pG, the Z=p-
cellularization of G is isomorphic to the fundamental group of ~PBZ=pBG, which *
*is
usually called DZ=pG.
Proof. Following ([RS98 ],3.3), the group DZ=pG is defined by a central extensi*
*on D
given by
0 -! H2(T Z=pG; Z[1=p]) -! DZ=pG -! TZ=pG -! 0
which is universal among the central extensions
0 -! B -! E -! TZ=pG -! 0
such that Hom (Z=p, B) = 0 and Ext(Z=p, B) = 0.
Now, these conditions hold for the extension E0 of 4.4 that defines CW Z=pG,
because p doesn't divide the order of A. So, there is just one map f : D -! E0
that is the identity over G.
On the other hand, as H2(T Z=pG; Z[1=p]) has no p-torsion it exists by 4.4 an
unique map g : E0 -! D which is again the identity over G. By universality,
E0' D and the result follows.
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS19
The last proposition states that the tools we have developed above for the c*
*ompu-
tation of the Z=p-cellularization of a group G remain useful for calculating DZ*
*=pG,
or in the language of group theory, the universal central extension of the Z=p-*
*radical
of G with coefficients in Z[1=p]. See ([RS98 ], 3.5) and ([MP01 ]) for more det*
*ails on
universal central extensions with coefficients.
We can also prove an analogous to the commutation result 3.1, which appears
as an easy consequence of the last statement.
Corollary 4.13. If G is a finite Z=p-cellular group , then there is an isomorph*
*ism
ß1~PBZ=pBG ' ~LZ=pG.
Proof. According to the previous proposition, the fundamental group of ~PBZ=pBG
is Z=p-cellular, so by 4.8 we obtain that H2(G; Z=p) = 0. This implies that DZ=*
*pG =
G, and G coincides with its Z=p-radical because it is Z=p-cellular. But the Z=*
*p-
radical is precisely the homotopy fiber of the Z=p-localization of G, so we have
finished.
It is worth to point out that the hypothesis of G Z=p-cellular is essential,*
* as you
can see taking for example G = Z=p2.
Once we have completed the description of the Z=p-cellularization of finite *
*groups,
we are prepared to present what is probably the main result of this section, a *
*com-
plete characterization of the finite groups G such that its classifying space B*
*G is
B Z=p-cellular.
Proposition 4.14. Let G be a finite Z=p-cellular group. Then BG is BZ=p-cellular
if and only if G is a p-group.
Proof. If G is not a p-group, Hn(G) is not p-torsion for a certain n 2. Using
the result ([RS98 ], 6.3), it is not M (Z=p, 1)-cellular for any two-dimensiona*
*l Moore
space M (Z=p, 1). Hence by 2.5 it cannot be BZ=p-cellular, because BZ=p is itse*
*lf
M (Z=p, 1)-cellular.
Conversely, suppose G is a p-group. We use induction over the order of the
group G. It is clear that B Z=p is B Z=p-cellular, so we admit the hypothesis *
*is
true for every group whose order is strictly smaller than pk. Let G be a group *
*of
order pk, then, and consider a minimal system of order p generators {x1, . .,.x*
*r, y},
that exists because the group is finite and Z=p-cellular. Denote by H the minim*
*al
subgroup of G generated by {x1, . .,.xr}. H is different from G, because the sy*
*stem
of generators is minimal, and in addition it is normal, because it is a maximal
subgroup of a nilpotent group. Now, it is easy to see that G=H is isomorphic to
Z=p and it is generated by the image of y. So, we can write G as a split extens*
*ion:
0 -! H -! G -! G=H -! 0.
By the induction hypothesis, BH is BZ=p-cellular, B(G=H) is BZ=p-cellular too
because it is isomorphic to Z=p, and as the extension is split, the associated *
*fibration
has a section. Then 2.5 implies that BG is BZ=p-cellular.
We will conclude this section by describing the fundamental group of CW BZ=*
*pBG.
Proposition 4.15. Let G be a finite Z=p-cellular group. Let us call r the order
of H2(G) and s the number of different homomorphisms Z=p -! G. Then the
fundamental group ß of the BZ=p-cellularization of BG fits in the following cen*
*tral
extension:
0 -! H -! ß -! G -! 0
20 RAM'ON J. FLORES
where ß is a finite p-group whose order is bounded by prs.
W
Proof. We consider again the evaluation map [BZ=p,BG]*-! BG, where the wedge
is extended over all the elements of [B Z=p, BG]*, and let Cf be again the homo*
*topy
cofibre of this map. The Mayer-Vietoris sequence of this cofibration is as foll*
*ows:
M
0 -! H2(B G) -! H2(Cf) -! Z=p -! Gab- ! 0
L
where the rank of the elementary abelian p-group Z=p is the number of homo-
morphisms Z=p -! G. The cofibre Cf is simply-connected, so H2(Cf) = ß2(Cf),
and its order is clearly bounded by prs. Now, the fundamental group ß is defined
by a central extension
0 -! A -! ß -! SZ=pG -! 0
where A is the second homotopy group of the B Z=p-nullification of Cf. Using t*
*he
fact that the suspension and Cf are simply-connected and the construction of the
nullification functor described in the preliminaries we obtain that A is a quot*
*ient
of ß2(Cf). We finish by observing that the Z=p-socle of G is Z=p-cellular and h*
*ence
equal to G.
The problem of determining exactly how many Z=p0s appear in the kernel of th*
*is
extension seems by no means easy stuff, because it depends essentially on how m*
*any
maps from M (Z=p, 2) to the succesive cofibers that appear in the construction *
*of
the B Z=p-nullification of Cf can be lifted to B Z=p. However, in next sectio*
*n we
will see some examples of groups for which is possible to determine exactly who*
* is
the fundamental group of CW BZ=pBG.
The previous result allows us to characterize the finite groups such that the
B Z=p-cellularization of its classifying space is a K(G, 1):
Proposition 4.16. Let G be a finite group. Then ßn(CW BZ=pBG) = 0 for n 2
if and only if the Z=p-cellularization of G is a p-group.
Proof. If CW Z=pG is a p-group, then by 4.14 and 4.3
CW BZ=pBG ' CW BZ=pBCW Z=pG ' BCW Z=pG.
Conversely, if CW BZ=pBG is an aspherical space BH, the group H is Z=p-cellula*
*r,
because taking fundamental group öc mmutes" with (homotopy) colimits. The
result follows now from 4.3 and the previous proposition.
5.Examples
In this section we will apply the theorem 3.3 for calculating explicitly the*
* BZ=p-
nullification and BZ=p-cellularization of the classifying spaces of some well-k*
*nown
finite groups. The omitted details of the structure of the groups involved can *
*be
found in [Wei77], [Gor80 ] or [Rob96 ]. We always suppose that the primes that
appear divide the order of G; otherwise classifying space of the group is BZ=p-*
*null
and hence its BZ=p-cellularization is a point.
5.1. Dihedral groups. Let Dn = {X, Y ; Xn = 1, Y 2= 1, (XY )2 = 1} be the
dihedral group of order 2n.
a) Nullification.
For computing PBZ=pBDn, we will distinguish the cases p 6= 2 and p = 2.
Suppose firstly that p is different of 2. Then |Dn| = prq, p coprime with q.*
* If
r = 0 the classifying space is BZ=p-null and we have nothing to say. Suppose r *
*> 0.
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS21
Now, the Z=p-radical T Z=pG of Dn is the subgroup generated by Xn=pr, which
is easily seen torbe the unique p-Sylow subgroup of Dn (inrparticular is normal*
*).
Moreover, < Xn=p > is isomorphic to Z=pr, and Dn=< Xn=p > is isomorphic to
Dn=pr. So, by the result 3.3 we have the following Postnikov fibration:
Y
(B Z=pr)^s-! PBZ=pBDn -! BDn=pr.
s6=p
Now, as (B Z=pr)^sis contractible if s 6= p we obtain PBZ=pBDn is homotopy
equivalent to BDn=pr.
Now we attack the case p = 2. It is clear by the relations that define the g*
*roup
that Y and XY belong to the Z=2-radical of Dn. ButQthis implies that X belongs
too, and then TZ=2(Dn) = Dn. Hence, PBZ=2BDn = q6=2(B Dn)^q.
b) Cellularization.
If p 6= 2, the Z=p-socle of Dn is the cyclic group of order p, whose generat*
*or is
identified inside Dn with Xn=p. Hence, the Z=p-cellularization of Dn is Z=p, and
then CW BZ=pBDn = BZ=p.
If p = 2, we can make the change Z = XY to obtain the presentation Dn =
{X, Z; X2 = 1, Z2 = 1, (XZ)n = 1}. This proves that Dn is always generated by
order two elements. In particular, if n = 2j for a certain natural number j, t*
*he
corresponding dihedral group is a 2-group, and hence by 4.14 its classifying sp*
*ace
is BZ=2-cellular.
In the case n 6= 2j, we can assure B Dn is not B Z=2-cellular, because it has
torsion in other prime, but we can prove that the group is Z=2-cellular. In fac*
*t, we
give an explicit construction of Dn, for every n, as a colimit of copies of Z=2.
Consider the second presentation given above, the usual presentations Z=2 =
{A; A2 = 1}, Z=2 * Z=2 = {B, C; B2 = 1, C2 = 1}, and suppose n is odd. Then Dn
is the coequalizer of the homomorphisms
Z=2 - ! Z=2 * Z=2
A - ! BCB . .C.B
where B appears n+1_2times, and
Z=2 -! Z=2 * Z=2
A -! CB . .B.C
where now C appears n+1_2times. The case n even is similar, with B appearing n_2
times in the first homomorphism, and C appearing n_2+ 1 times in the second. As
every coequalizer is a colimit, we have proved that the dihedral groups are alw*
*ays
Z=2-cellular.
5.2. Finite projective special linear groups and special linear groups. Next
we study the family of groups SL(2, q), q prime, and its quotient groups P SL(2*
*, q).
If q = 2, then SL(2, q) = P SL(2, q) = D6, and this case has been already studi*
*ed
in the previous section. Rememeber in the sequel that P SL(2, q) is simple if q*
* 5.
a) Nullification.
Consider the presentation of the tetrahedral group P SL(2, 3) given by {X, Y*
* ; X3 =
1, Y 3= 1, (XY )2 = 1}. The unique normal subgroup of P SL(2, 3) that is not tr*
*ivial
is H = {1, XY, Y X, XY 2X}, which is isomorphic to the Klein group Z=2xZ=2, and
it is easy to see that this is precisely the Z=2-radical of P SL(2, 3). The ass*
*ociated
extension gives rise to a fibration of classifying spaces
BZ=2 x BZ=2 -! BP SL(2, 3) -! BZ=3.
As the fibre is BZ=2-acyclic, we have a homotopy equivalence PBZ=2BP SL(2, 3) '
B Z=3. For p = 3, P SL(2, 3) is generated by order 3 elements, so it is equal
22 RAM'ON J. FLORES
to its Z=3-radical, and hence PBZ=3BP SL(2, 3) ' B P SL(2, 3)^2. If q 5, then
T Z=p(P SL(2, q)) = P SL(2, q) for every prime p dividing the order of P SL(2, *
*q),
because the group is simple. Then, by 3.3, the BZ=p-nullification of BP SL(2, *
*q)
is homotopy equivalent to the Z[1=p]-completion of BP SL(2, q). We consider now
the non-projective case. We have always the fibration
BZ=2 -! BSL(2, q) -! BP SL(2, q).
Now the base is BZ=2-acyclic, and we have PBZ=2BP SL(2, q) ' PBZ=2BSL(2, q),
which is again simply connected except for the pathological case q = 3. If p 6=*
* 2 and
divides the order of P SL(2, q), this group is always generated by the transvec*
*tions
of order p ([Rob96 ],3.2.10) and again the BZ=p-nullification of its classifyin*
*g space
is homotopy equivalent to its Z[1=p]-completion.
b) Cellularization.
We will again consider first the case q = 3, the tetrahedral group.
If p = 2, we have said that the Z=2-radical is the Klein group Z=2 x Z=2, wh*
*ich
is equal to its Z=2-socle, because it is generated by order 2 elements. As the *
*Klein
group is indeed a 2-group, we have CW BZ=2BP SL(2, 3) = BZ=2 x BZ=2.
In the case p = 3, recall P SL(2, 3) is generated by the order 3 transvectio*
*ns, so
it is equal to its Z=3-socle. It is easy to see that the Schur multiplier H2(P *
*SL(2, 3))
is Z=2, and then according to 4.4, the Z=3-cellularization of P SL(2, 3) is def*
*ined
by a central extension
0 -! Z=2 -! CW Z=3P SL(2, 3) -! P SL(2, q) -! 0.
This extension is nontrivial by 4.11, and it is known that the only nontrivial *
*central
extension of P SL(2, 3) by Z=2 is SL(2, 3), so the Z=3-cellularization of P SL(*
*2, 3)
is SL(2, 3) and hence the latter group is Z=3-cellular, because cellularization*
* is an
idempotent functor.
If q 5, the group P SL(2, 3) is perfect, and its universal central extensi*
*on is
precisely
0 -! Z=2 -! SL(2, 3) -! P SL(2, q) -! 0.
Hence, the Schur multiplier of P SL(2, q) is Z=2, and by 4.7 this group is Z=2-
cellular. On the other hand, SL(2, q) is the universal covering group of itself*
*, and
in particular H2(SL(2, q)) = 0, so by corollary 4.10 it is Z=2-cellular. For p *
*odd, the
same corollary proves that if p divides the order of P SL(2, 3), the Z=p-cellul*
*arization
of P SL(2, q) is SL(2, q), and this group is Z=p-cellular.
To conclude, observe that neither SL(2, q) nor P SL(2, q) are p-groups for a*
*ny
primes p 2 and q > 2; hence, its classifying spaces cannot be B Z=p-cellular.
It is interesting to point out that B SL(2, q) cannot be the B Z=p-cellularizat*
*ion
of BP SL(2, q), not even in the case p odd, in which we know SL(2, q) to be the
Z=p-cellularization of P SL(2, q).
5.3. Symmetric and alternating groups. In this section we will put our atten-
tion over the permutation group Sn and the alternating group An, the subgroup of
permutations of even signature of Sn.
a) Nullification.
The symmetric group of n-letters (n 2) admits the following presentation:
Sn = {X1, . .,.Xn-1; XiXi+1Xi= Xi+1XiXi+1, XiXi+jXi= Xi+j forj 2}.
This shows that Sn is generated by order 2 elements, and in particular is equal*
* to
its Z=2-radical. Hence, its BZ=2-nullification is homotopy equivalent to its Z[*
*1=2]-
completion. If p is odd, we can consider the fibration defined by the inclusion*
* of
the alternating group:
NULLIFICATION AND CELLULARIZATION OF CLASSIFYING SPACES OF FINITE GROUPS23
B An -! BSn -! BZ=2.
As the base is BZ=p-null for p odd, the fibration is preserved under BZ=p-nulli*
*fication.
If n = 3, A3 = Z=3, and then PBZ=3BS3 ' BZ=2. In the case n 4, An is always
generated by order p elements: if n = 4, it is known that A4 = P SL(2, 3), and
hence is generated by the 3-transvections, as we have said before; if n > 4, An*
* is
simple, and then the Z=p-radical, which is normal, is the whole group. So, these
considerations prove that for n 4 and p odd, PBZ=pBAn is homotopy equivalent
to Z[1=p]1 BAn. In particular, the fibration which defines the BZ=p-nullificati*
*on of
B Sn takes the form
Z[1=p]1 BAn -! PBZ=pBSn -! BZ=2
which turns to be a Postnikov fibration, because Z[1=p]1 BAn is simply connecte*
*d.
The previous arguments also prove that PBZ=2BAn ' Z[1=2]1 BAn if n 5. If
n = 4 the alternating group is isomorphic to the tetrahedral group and this case
have been studied in the previous section.
b) Cellularization.
In ([RS98 ], 6.5) Rodr'iguez-Scherer build effectively Sn as a colimit of co*
*pies of
Z=2, so Sn is Z=2-cellular. If p is odd, let us see that every permutation of o*
*rder p
is even. Let oe be such a permutation. Then
1 = sig(Id) = sig(oep) = (sig(oe))p
and, if p is odd, this is only possible if sig(oe) = 1, what amounts to say tha*
*t oe is even.
This means that the Z=p-socle of Sn is a subgroup of An, which is normal because
the socle is always a characteristic subgroup; this already proves that SZ=pSn *
*= An,
and in particular CW Z=3S4 = CW Z=3A4 = SL(2, 3). So, we fix our attention again
in alternating groups An with n 5. It is known ([Asc00], 33.15) that the Schur
multiplier of An is Z=2 for every n 6= 6, 7, and H2(A6) = H2(A7) = Z=6. So, if
p 5, 4.10 imply that the Z=p-cellularization of An is in this case isomorphic*
* to the
universal central extension of An. If p = 3, according to 4.11 the Z=3-cellular*
*ization
is given by the unique nontrivial extension of An by Z=2, and finally, if p = 2*
*, An
is Z=2-cellular for every n different from 6 or 7; in the pathological case, 4.*
*9 tells us
that the Z=2-cellularization of An for n = 6, 7 is the extension of An by Z=3 t*
*hat
corresponds to element of H2(An; F3) that goes to the identity via the isomorph*
*ism
H2(An; F3) ' Hom (F3, F3).
As the order of Sn and An is respectively n! and n!=2, these groups are never
p-groups for n 2, and hence their classifying spaces cannot be BZ=p-cellular.
5.4. p-groups. We finish by describing the effect of the BZ=p-cellularization f*
*unc-
tor on the classifying spaces of three family of p-groups, namely the quaternio*
*nic
groups, semidihedral groups and Mm (p)-groups.
Consider the quaternion group Qm+1 = {H, K; H2m-1 = K2 = 1, HK =
KH-1 }, with m 3. This group has order 2m+1 , and the Z=2-socle is the center,
which is the subgroup generated by H2m-1. This subgroup is isomorphic to Z=2,
and hence CW BZ=2BQm ' BZ=2 for every m.
We turn now our attention to themsemidihedral-groups1of orderm2m-,2that admit
a presentation SDm = {X, Y ; X2 = Y 2= 1, Y XY -1 = X-1+2 }. The
Z=2-socle of SDm is generated by X2m-2 and Y , and it is isomorphic to D2m-2.
So, CW BZ=2BSDm is homotopy equivalent to BD2m-2.
Finally, if p = 2mand-m1> 3, or if p is odd andmm->22, we define the group
Mm (p) = {X, Y ; Xp = Y p = 1, Y XY -1 = X1+p }. The Z=p-socle of
24 RAM'ON J. FLORES
Mm (p) is generated by Xpm-2 and Y and it is isomorphic to Z=p x Z=p. Thus, the
B Z=p-cellularization of BMm (p) is BZ=p x BZ=p.
These last computations have been very simple, but they give us an explicit
expression for the B Z=p-cellularization of a very wide class of p-groups, among
which we quote the following families:
o The p-groups of order pm which contain a cyclic subgroup of order pm-1 .
o The p-groups with no noncyclic abelian normal subgroups.
o The groups of order p3.
The proof of this relies on the classifying results that can be found, for e*
*xample,
in ([Gor80 ], 5.4).
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Departamento de Matem'aticas, Universidad Aut'onoma de Barcelona, E-08193 Be*
*l-
laterra, Spain
E-mail address: ramonj@mat.uab.es