Essential Cohomology of Finite Groups
Alejandro Adem * and Dikran Karagueuzian**
Mathematics Department
University of Wisconsin
Madison, WI 53706 USA
x0 Introduction
One of the main difficulties in calculating or understanding the mod p coho*
*mology
of a finite group G is the fact that in many instances non-trivial cohomology c*
*lasses
will appear that restrict trivially on all proper subgroups. Such "essential" *
*cohomology
(denoted Ess*(G)) is difficult to obtain and yet seems to be an intrinsic contr*
*ibution of
some importance. On the other hand, it is evident from the definition that such*
* groups
will be "universal detectors" in the cohomology of finite groups. However, this*
* detection
scheme is much too general to be of any practical significance. Rather, one wou*
*ld hope
that a certain accessible and well-understood collection of these groups would *
*suffice under
favorable conditions. In particular a group-theoretic characterization of such *
*groups would
be highly desirable.
In this paper we obtain such a result for certain groups, namely those whos*
*e cohomol-
ogy ring is Cohen-Macaulay (i.e. free and finitely generated over a polynomial *
*subalgebra).
More precisely, we prove the following
Theorem 2.1:
Let G be a finite group, then the following two conditions are equivalent:
(1) H*(G; Fp) is Cohen-Macaulay and contains non-trivial essential elements
(2) G is a p-group and every element of order p in G is central. |
We note that our theorem is a long-sought cohomological characterization of*
* this
particular type of finite p-group (which we shall say satisfies the pC conditio*
*n). Such
results are not common in the cohomology of finite groups, as the structure of *
*a group
is usually far more rigid than its cohomology. An interesting consequence is t*
*hat given
a group with Cohen-Macaulay cohomology and containing essential elements, the s*
*ame
property will be inherited by any subgroup. Combining the above with a detectio*
*n theorem
due to J. Carlson [C], we obtain
_________________________
* Partially supported by an NSF grant.
** Supported by an NSF Postdoctoral Fellowship.
1
Corollary 2.2:
Let Q be a finite group with G 2 Sylp(Q); then, if E G Q is an elementary
abelian subgroup of maximal rank, H*(CG (E); Fp) is Cohen-Macaulay and has non-*
*trivial
essential elements. Moreover, if H*(Q; Fp) is Cohen-Macaulay, then it is detect*
*ed on the
cohomology of these centralizers. |
Hence we see that in the Cohen-Macaulay situation, there is universal detec*
*tion by
accessible groups with non-trivial essential cohomology; the usefulness of such*
* a result is
illustrated for example in the calculations of the mod 2 cohomology of the spor*
*adic simple
groups M11, M12 and O0N. We will discuss these examples in x3, as well as the s*
*ituation
for lower depth, using M22 as an example.
We should point out that this paper developed as an attempt to settle probl*
*em 4 in
J.F. Adams' "Problem Session for Homotopy Theory" held at the 1986 Arcata Topol*
*ogy
Conference (see [CCMR], page 438): namely can one give a useful alternative des*
*cription
of p-groups with non-zero essential cohomology? M. Feshbach's original conject*
*ure was
that Ess*(G) 6= 0 if and only if G satisfies the pC condition (restricted to p *
*= 2). Rusin [R]
provided an example (denoted 6a2 in [H]) of a group of order 32 such that Ess*(*
*G) 6= 0
but without the 2C condition. The extra-special p-groups of order p3 and expon*
*ent p,
p an odd prime greater than 3, are counterexamples for p odd (see [L] and x3). *
*In both
instances it is apparent that the Cohen-Macaulay condition is absent. However,*
* by a
result due to Duflot [D], the pC condition implies Cohen-Macaulay, hence the mo*
*tivation
behind Theorem 2.1.
The main idea in our proof is to exploit the current technical understandin*
*g of Cohen-
Macaulay cohomology rings, as developed by Benson and Carlson in [BC]. The unde*
*rlying
explicit `geometric' structure which they develop provides the groundwork for a*
* construc-
tive method used in proving the existence of undetectable cohomology classes gi*
*ven the
pC hypothesis.
The paper is organized as follows: in x1 we provide background in concepts*
* and
methods from group cohomology, in x2 we prove our main results and in x3 we des*
*cribe
some examples and draw a few conclusions. We are indebted to D. Benson and M. I*
*saacs
for helpful comments.
x1 Preliminaries
In this section we will provide the background necessary for the proof of o*
*ur main
results. The books [B] and [AM1] are good general references. We assume throu*
*ghout
(unless otherwise stated) that G is a finite p-group and that k is a field of c*
*haracteristic
p. To begin we recall
Definition 1.1:
The cohomology ring H*(G; k) is said to be Cohen-Macaulay if there exists a*
* poly-
nomial subalgebra
k[i1; : :;:in] H*(G; k)
2
over which H*(G; k) is a free and finitely generated module. |
This is an important condition on the cohomology of a finite group which we*
* shall
use decisively in the proof of our main result. We will also need the following*
* basic result
due to J. Carlson (see [C], Theorem 2.3)
Theorem 1.2:
Let G be a finite group such that H*(G; k) is Cohen-Macaulay. Then its coho*
*mology
is detected by restriction to the cohomology of centralizers of elementary abel*
*ian subgroups
of maximal rank in G. |
Next we introduce
Definition 1.3:
The group G is said to satisfy the pC condition if every element of order p*
* in G is
central. |
Well known examples of this type of group include abelian p-groups and quat*
*ernion
groups. Less well known is the fact that the 2-Sylow subgroups of the simple g*
*roups
P SU3(F2n) and Sz(22n+1) also satisfy this condition.
The two concepts above are related by a result due to J. Duflot [D]
Theorem 1.4:
Let E G be an elementary abelian subgroup of maximal rank in G; then, if C*
*G (E)
denotes its centralizer in G, the cohomology ring H*(CG (E); k) is Cohen-Macaul*
*ay. |
Corollary 1.5:
If G satisfies the pC condition, then H*(G; k) is Cohen-Macaulay. |
Indeed, if G satisfies the pC condition, then it has a unique central eleme*
*ntary abelian
subgroup of maximal rank.
We now introduce another cohomological notion.
Definition 1.6:
An element x 2 H*(G; k) is said to be essential if it restricts to zero on *
*every proper
subgroup in G. |
If (G) denotes the collection of all index p subgroups in G, then we can co*
*nsider the
map induced by restrictions
M
H*(G; k) -! H*(H; k):
H2(G)
3
The essential elements are precisely those in the kernel of this map. Let us d*
*enote the
ideal of essential elements by Ess*(G) H*(G; k). We record without proof some *
*basic
properties of this object.
Proposition 1.7:
(1) Ess*(G) = 0 if and only if H*(G; k) is detected on proper subgroups. In*
* particular
it is zero if G is not a p-group.
(2) Ess*(G) is an ideal invariant under the action of the Steenrod Algebra.
(3) If G is not elementary abelian, the elements in Ess*(G) are nilpotent.
(4) If f : G ! K is a group isomorphism, then it induces a bijection Ess*(K*
*) !
Ess*(G). |
It is clear that there exist many interesting groups for which Ess*(G) = 0.*
* In fact
this vanishing is a key element in most known cohomology calculations (see [AM1*
*] for
explicit examples). For example, it has long been understood that detectability*
* on abelian
subgroups is a highly desirable situation that arises in a number of important *
*cases. In
contrast, a group G with non-zero essential elements in its cohomology contribu*
*tes new
information inherently associated to the group. In turn it will play the part o*
*f detector
for groups which contain it. Hence it is natural to search for group-theoretic *
*conditions
which imply Ess*(G) 6= 0.
x2 Main Result
In this section we prove our main result, namely
Theorem 2.1:
Let G be a finite p-group. Then every element of order p in G is central if*
* and only
if H*(G; k) is Cohen-Macaulay and Ess*(G) 6= 0.
Proof: Assume first that every element of order p in G is central. By 1.5, this*
* hypothesis
implies that H*(G; k) is Cohen-Macaulay. Hence we may choose elements
i1; : :;:in 2 H*(G; k)
such that they generate a polynomial subalgebra
k[i1; : :;:in] H*(G; k)
over which H*(G; k) is free and finitely generated. Note that n is equal to the*
* p-rank of
G, the rank of the largest elementary abelian subgroup in G.
Using theorems 1.3 and 6.3 in [BC], it follows that we may choose a free ge*
*nerator
z 2 HN (G; k) for H*(G; k) as a module over k[i1; : :;:in] such that z = trGE*
*(x), where
E G is the central elementary abelian subgroup in G of maximal rank (this elem*
*ent is
referred to as the "last survivor").
4
Now by Quillen's theorem on nilpotent elements (see [AM1]), restriction to *
*H*(E; k)
must be injective on k[i1; : :;:in]. Let i0i= resGE(ii), then
k[i01; : :;:i0n] H*(E; k)
is a polynomial subalgebra of maximum Krull dimension n. If (E) denotes the col*
*lection
of all index p subgroups in E, then the map induced by restrictions
M
k[i01; : :;:i0n] -! H*(E0; k)
E02(E)
must have a non-trivial kernel (as can be seen by comparing Krull dimensions). *
*Hence we
may choose a non-zero polynomial
p(i01; : :;:i0n) 2 Ess*(E) \ k[i01; : :;:i0n]:
We now take the element
p(i01; : :;:i0n) . x 2 H*(E; k)
and apply trGEto it:
trGE(p(i01; : :;:i0n)=. x)trGE(resGE(p(i1; : :;:in)) . x)
= p(i1; : :;:in) . trGE(x)
= p(i1; : :;:in) . z 6= 0:
We claim that this element is essential. Let H G be any index p subgroup i*
*n G,
then using the double coset formula (see [AM1], II.6), we obtain
resGH[p(i1; : :;:in)=. z]resGHtrGE[p(i01; : :;:i0n) . x]
X g
= trHH\EgresEH\Eg[g . p(i01; : :;:i0n) . x*
*]:
[g]2H\G=E
Note however that E G is central, hence Eg = E and the action in cohomology *
*is
trivial. Hence the expression above simplifies to yield
X
resGH[p(i1; : :;:in) . z] = trHH\E resEH\E[p(i01; : :;:i0n)*
* . x]:
[g]2H\G=E
We now consider the two possible cases: either E H or not. Assume to begin*
* that
H \ E 6= E; in this case we have
resEE\H[p(i01; : :;:i0n) . x] = resEH\E(p(i01; : :;:i0n)) . resEH\E*
*x = 0
5
as p(i01; : :;:i0n) 2 H*(E; k) is essential by construction. Hence we conclude *
*that
resGH[p(i1; : :;:in) . z] = 0:
Now let's assume that H \ E = E; in this case the formula simplifies to
X
resGH[p(i1; : :;:in).z] = trHH\E[p(i01; : :;:i0n).x] = [G : H]trHH\E[p(i01*
*; : :;:i0n.x] = 0:
[G:H]
We conclude that in either case the restriction is zero, and hence p(i1; : *
*:;:in) . z is a
non-zero essential element and so Ess*(G) 6= 0.
For the converse we use the following interpretation of 1.2: if there exist*
*s an element
6= 0 in H*(G; k) such that resGCG(E)() = 0 for all elementary abelian subgrou*
*ps E of
maximal rank with centralizer CG (E), then H*(G; k) is not Cohen-Macaulay.
Hence if Ess*(G) 6= 0 and H*(G; k) is Cohen-Macaulay, we conclude that th*
*ere
exists an elementary abelian subgroup E G of maximal rank such that G = CG (*
*E).
This clearly implies that all elements of order p in G are central, and therefo*
*re our proof
is complete. |
Remark: Let G be any p-group of rank n. Then, if E G is elementary abelian of
this rank, we have shown that CG (E) has essential elements in its cohomology, *
*which in
addition is Cohen-Macaulay. If we take Q to be any finite group with G 2 Sylp(Q*
*), then
1.2 will still hold for Q. Noting that (up to conjugacy) we may assume Sylp(CQ *
*(E)) =
G \ CQ (E) = CG (E), we can deduce
Corollary 2.2:
Let Q be a finite group with G 2 Sylp(Q); then, if E G Q is an elementary
abelian subgroup of maximal rank, H*(CG (E); k) is Cohen-Macaulay and has non-t*
*rivial
essential elements. Moreover, if H*(Q; k) is Cohen-Macaulay, then it is detecte*
*d on the
cohomology of these subgroups. |
A simple but nevertheless interesting by-product of our theorem is the foll*
*owing
Corollary 2.3:
If H*(G; k) is Cohen Macaulay and Ess*(G) 6= 0, then the same will hold f*
*or any
subgroup of G. Similarly if p is an odd prime and E is the maximal elementary a*
*belian
subgroup in G, the same properties will hold for G=E.
Proof: The proof for subgroups follows from the simple observation that the pC *
*condition
is inherited on subgroups. Less well known is the fact (pointed out to us by M*
*. Isaacs
[I]) that for p an odd prime, if G is pC and E denotes the maximal elementary a*
*belian
subgroup, then G=E will also be pC. This completes the proof. |
6
We already mentioned in x1 that from Quillen's work it follows that all the*
* elements
in Ess*(G) are nilpotent provided G is not elementary abelian. The following pr*
*oposition
shows how they appear in the cohomology of groups Q such that Sylp(Q) = G.
Proposition 2.4:
Let G be a p-Sylow subgroup of Q, then im resQG\Ess*(G) = Ess*(G)NQ(G), whe*
*re
NG (Q) denotes the normalizer of G in Q. These invariant classes represent elem*
*ents in the
cohomology of Q and in particular the elements in Ess*(G)Out(G) are universally*
* stable,
i.e. will be in the restriction image for any Q with G as a p-Sylow subgroup.
Proof: Let x 2 Q; we begin by observing that the map induced by conjugating wit*
*h x
will induce an isomorphism c*x: Ess*(G) ! Ess*(Gx) (this follows from 1.7).
Now let Q = tni=1GxiG be a double coset decomposition. By the Cartan-Eilenb*
*erg
stability theorem (using the fact that [Q : G] is prime to p) an element z 2 H**
*(G; k) is in
the image of resQGif and only if it is stable, i.e.
xi *
resGG\Gxi(z) = resGG\Gxi. cxi(z)
for i = 1; 2; : :;:n (see [AM], II.6). If z 2 Ess*(G) and Gxi 6= G, then both s*
*ides are zero
and stability follows. If xi 2 NQ (G), then the condition simply means that c*x*
*i(z) = z
(note that if OE 2 Aut(G), then by 1.7 it induces a bijection on Ess*(G)). We *
*conclude
that Ess*(G)NQ(G) = im resQG. Finally we note that inner automorphisms act tr*
*ivially
in cohomology and that
Ess*(G)Out(G) \Q Ess*(G)NQ(G)
where the term on the right is the set of universally stable elements in Ess*(G*
*). |
Remark: In [MP] it is shown that if G satisfies the pC condition (p odd), then *
*H*(Q; k) ~=
H*(G; k)NQ(G) for any Q such that Sylp(Q) = G. For p = 2, it is possible to s*
*how that
H*(G; k) ~=H*(NQ (E); k), where E G is the maximal elementary abelian subgroup.
x3 Examples and Conclusions
The results in this paper indicate that groups of type pC will be a detecti*
*ng family with
non-zero essential cohomology for all groups which satisfy the Cohen-Macaulay c*
*ondition
on their cohomology. We present a few interesting examples.
Example 3.1: Let M11 denote the first Mathieu group. Then its 2-Sylow subgroup *
*is
semi-dihedral of order 16, with a unique maximal elementary abelian subgroup of*
* rank 2
which is self-centralizing. H*(M11; F2) is Cohen-Macaulay, detected on this sub*
*group (see
[AM1]). Note that in contrast Syl2(M11) is not Cohen-Macaulay and is not detect*
*ed on
this elementary abelian subgroup.
Example 3.2: Let M12 denote the next Mathieu group; its 2-Sylow subgroup has or*
*der
64, a semi-direct product Q8 xT D8 and there are three conjugacy classes of max*
*imal
7
rank (equal to 3) elementary abelian subgroups. They are all self-centralizing.*
* Hence, as
H*(M12; F2) is Cohen-Macaulay, they will detect the cohomology (see [AM1]).
Example 3.3: Let O0N denote the O'Nan group; its 2-Sylow subgroup has order 29,
a nonsplit extension (Z=4)3 . D8, and there are exactly 2 conjugacy classes of *
*maximal
rank (equal to 3) elementary abelian subgroups. One has centralizer (Z=4)3, the*
* other has
centralizer Z=2 x Z=2 x Z=4. Once again, H*(O0N; F2) is Cohen-Macaulay and they*
* must
detect the cohomology (see [AM2]).
The converse of our main theorem is actually a special consequence of a mor*
*e general
result in [C]. Recall that the depth of H*(G; k) is the longest length of a seq*
*uence of regular
elements. Then 1.2 extends to the following statement: if depth H*(G; k) d, *
*then
H*(G; k) is detected on the cohomology of centralizers of elementary abelian su*
*bgroups
in G of rank d. In particular as before we conclude
Proposition 3.4:
If Ess*(G) 6= 0, and H*(G; k) has depth d, then G centralizes an elementary*
* abelian
subgroup of rank d. |
A plausible next step in characterizing groups with essential cohomology wo*
*uld be to
find necessary and sufficient group-theoretic conditions on G so that depth H*(*
*G; k) = d,
where d is less than the rank of G, and Ess*(G) 6= 0.
Examples of such types of groups include the extra-special p-groups Gp of o*
*rder p3
and exponent p, where p > 3 (for p = 3 the mod p cohomology is Cohen-Macaulay a*
*nd
is detected on proper subgroups [MT]). The cohomology H*(Gp; Fp) has depth one *
*and
Ess*(Gp) 6= 0 (see [L]). And of course Gp does not centralize any rank 2 elem*
*entary
abelian subgroup. The following examples illustrate a 2-stage detection scheme *
*without
the Cohen-Macaulay condition.
Example 3.5: Consider the projective matrix group L3(4); its 2-Sylow subgroup S*
* is of
order 64, a central extension of (Z=2)4 by (Z=2)2. An explicit calculation [AM3*
*] shows that
H*(S) is not Cohen-Macaulay; however it is detected on subgroups of the form (Z*
*=2)4
and Q8 x Z=2, which certainly satisfy the 2C condition (we are grateful to Jim *
*Milgram
for pointing out this detection scheme to us).
Example 3.6: Let M22 denote the next Mathieu group after M12, its 2-Sylow subgr*
*oup
has order 27, a semi-direct product (Z=2)4xT D8. This simple group has 3 conjug*
*acy classes
of maximal elementary abelian subgroups: two of rank 4 and one of rank 3; they *
*are all
self-centralizing. We deduce that H*(M22; F2) is not Cohen-Macaulay. The cohomo*
*logy
is detected by its elementary abelian subgroups together with an index two subg*
*roup
R ~= Syl2(L3(4)) (see [AM4]). Combining this with the previous detection state*
*ment,
we conclude that H*(M22; F2) is detected on elementary abelian subgroups togeth*
*er with
subgroups of the form Q8 x Z=2.
This last example illustrates how essential cohomology on detecting subgrou*
*ps can
build up cohomology in a complicated group regardless of the Cohen-Macaulay con*
*dition.
8
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*y of
O'Nan's Sporadic Simple Group," J. Algebra 176 (1995), 288-315.
[AM3] Adem, A. and Milgram R.J., "A5-invariants, the Cohomology of L3(4) and*
* Related
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9