On Commuting and Non-Commuting Complexes
Jonathan Pakianathan and Erg"un Yalcin
April 19, 2000
Abstract
In this paper we study various simplicial complexes associated to the
commutative structure of a finite group G. We define NC(G) (resp. C(G))
as the complex associated to the poset of pairwise non-commuting (resp.
commuting) sets of nontrivial elements in G.
We observe that NC(G) has only one positive dimensional connected
component, which we call BNC(G), and we prove that BNC(G) is simply
connected.
Our main result is a simplicial decomposition formula for BNC(G)
which follows from a result of A. Bj"orner, M. Wachs and V. Welker on
inflated simplicial complexes (See [BWW ].) As a corollary we obtain
that if G has a nontrivial center or if G has odd order, then the homology
group Hn-1(BNC(G)) is nontrivial for every n such that G has a maximal
noncommuting set of order n.
We discuss the duality between NC(G) and C(G), and between their
p-local versions NCp(G) and Cp(G). We observe that Cp(G) is homotopy
equivalent to the Quillen complexes Ap(G), and obtain some interesting
results for NCp(G) using this duality.
Finally, we study the family of groups where the commutative relation
is transitive, and show that in this case, BNC(G) is shellable. As a
consequence we derive some group theoretical formulas for the orders of
maximal non-commuting sets.
1991 Mathematics Subject Classification. Primary: 20J05; Secondary:
06A09, 05E25.
1 Introduction
Given a finite group G, one defines a non-commuting set to be a set of elements
{g1; : :;:gn} such that gi does not commute with gj for i 6= j. The sizes of
maximal non-commuting sets in a group are interesting invariants of the group.
In particular, the largest integer n such that the group G has a non-commuting
set of order n, which is denoted by nc(G), is known to be closely related to
other invariants of G. For example if k(G) is the size of the largest conjugacy
class in G then
k(G) 4(nc(G))2 (See [P ].)
1
Also if we define cc(G) to be the minimal number of abelian subgroups of G
that covers G, then I. M. Isaacs (see [J]) has shown that
nc(G) cc(G) (nc(G)!)2:
Confirming a conjecture of P. Erd"os, L. Pyber [P ] has also shown that there
is a positive constant c such that
cc(G) |G : Z(G)| cnc(G)
for all groups G. Another interesting place where the invariant nc(G) appears
is in the computation of the cohomology length of extra-special p-groups (See
[Y ].)
In this paper we study the topology of certain complexes associated to the
poset of non-commuting sets in a group G. Let NC(G) be the complex whose
vertices are just the nontrivial elements of the group G and whose faces are the
noncommuting sets in G. The central elements form point components in this
complex and are not so interesting. So, we look at the subcomplex BNC(G)
formed by non-central elements of G. We show
Result 1 ( 3.2).If G is a nonabelian group, then BNC(G) is simply-connected.
In general we also notice that BNC(G) is equipped with a free Z(G)-action
where Z(G) is the center of G. It is also equipped in general with a Z=2Z-action
whose fixed point set is exactly BNC2(G), the corresponding complex where we
use only the elements of order 2 (involutions). Thus if G is an odd order group
or if G has nontrivial center, then the Euler characteristic of BNC(G) is not 1
and so it is not contractible.
On a more refined level, we use a recent simplicial decomposition result of *
*A.
Bj"orner, M. Wachs and V. Welker to show that there is a simplicial complex S,
called the core of BNC(G), so that the following decomposition formula holds:
Result 2 ( 4.11). If G is a finite nonabelian group and S is the core of BNC(G),
then _
BNC(G) ' S _ [Susp|F|(LkF )]fl(F)
F2S
where the F are the faces of S, Lk stands for the link of a face, Suspk stands
forQa k-fold suspension and _ stands for wedge product. The number fl(F ) =
[x]F (mx - 1) where mx is the size of the centralizer class [x].
It is clear from this decomposition formula that when the core of BNC(G)
is contractible, then BNC(G) is a wedge of suspensions of spaces and hence
has a trivial ring structure on its cohomology. This is true, for example, when
G = p, the symmetric group on p letters, for some prime p.
The following is an important consequence of the above decomposition for-
mula:
2
Result 3 ( 4.14). Let G be a finite nonabelian group, and let Ss denote the set
of maximal non-commuting sets in G of size s. Then, for s > 1,
X Y 1
rk(Hs-1(BNC(G))) [ (1 - ___)]
F2Ss x2F mx
where mx is the size of the centralizer class containing x.
In particular, if G is an odd order group or if G has a nontrivial center
(|G| 6= 2), then
H"s-1(NC(G)) 6= 0
whenever G has a maximal non-commuting set of size s.
There is also a p-local version of this theorem which, in particular, gives *
*that
H"s-1(NCp(G)) 6= 0 whenever G has a maximal non-commuting p-set of size s
and p is an odd prime. For p = 2, the same is true under the condition 2||Z(G)|
and |G| 6= 2. Observe that this result has a striking formal similarity (in ter*
*ms
of their conclusions) to the following theorem proved by Quillen (theorem 12.1
in [Q ]):
Theorem 1.1 (Quillen). If G is a finite solvable group having no nontrivial
normal p-subgroup, then
"Hs-1(Ap(G)) 6= 0
whenever G has a maximal elementary abelian p-group of rank s.
A nice consequence of Result 3 can be stated as follows: If G is a group of
odd order or a group with nontrivial center such that BNC(G) is spherical, i.e.,
homotopy equivalent to a wedge of equal dimensional spheres, then all maximal
non-commuting sets in G have the same size.
A natural question to ask is: For which groups is BNC(G) spherical? As
a partial answer, we show that if G is a group where the commutation relation
is transitive, then BNC(G) is spherical. We give examples of such groups and
compute BNC(G) for these groups.
Observe that one could also define a commuting complex C(G) analogous
to the way we defined NC(G) by making the faces consist of commuting sets
of elements instead of non-commuting sets. However, this definition does not
provide us with new complexes. For example, Cp(G), the commuting complex
formed by the elements of prime order p, is easily shown to be G-homotopy
equivalent to Quillen's complex Ap(G). However, the definition of Cp(G) helps
us to see a duality between NCp(G) and Cp(G) where NCp(G) is the subcomplex
of NC(G) spanned by the vertices which correspond to elements of order p.
Using a result of Quillen on Ap(G), we obtain:
Result 4 ( 5.2).Let G be a group and p a prime with p||G|. Pick a Sylow p-
subgroup P of G and define N to be the subgroup generated by the normalizers
NG (H) where H runs over all the nontrivial subgroups of P .
Then NCp(G) is (|G : N|-2)-connected. In fact NCp(G) is the |G : N|-fold
join of a complex S with itself where S is "dual" to a path-component of Ap(G).
3
Finally under suitable conditions, BNC(G) is shellable and this yields some
combinatorial identities. As an application we obtain
Result 5 ( 4.24). Let G be a nonabelian group with a transitive commuting
relation, i.e., if [g; h] = [h; k] = 1, then [g; h] = 1 for every noncentral g;*
* h; k 2
G. Then,
nc(G)(nc(G) - 1) + |G|(|G| - m) - 2(nc(G) - 1)(|G| - |Z(G)|) 0:
where m denotes the number of conjugacy classes in G.
2 Background
We start the section with a discussion of complexes associated with posets of
subgroups of a group G. For a complete account of these well known results,
see chapter 6 in [B ].
Given a finite poset (P; ), one can construct a simplicial complex |P | out
of it by defining the n-simplices of |P | to be chains in P of the form p0 < p1*
* <
. .<.pn. This is called the simplicial realization of the poset P .
Furthermore any map of posets f : (P1; 1) ! (P2; 2) (map of posets
means x1 1 x2 ) f(x1) 2 f(x2)) yields a simplicial map between |P1| and
|P2| and hence one has in general a covariant functor from the category of fini*
*te
posets to the category of finite simplicial complexes and simplicial maps. Thus
if a (finite) group G acts on a poset P via poset maps (we say P is a G-poset)
then G will act on |P | simplicially.
Brown, Quillen, Webb, Bouc, Thevenaz and many others constructed many
finite G-simplicial complexes associated to a group G and used them to study
the group G and its cohomology. In particular, the following posets of subgroups
of G have been studied extensively:
(a) The poset sp(G) of nontrivial p-subgroups of G.
(b) The poset ap(G) of nontrivial elementary abelian p-subgroups of G.
(c) The poset bp(G) of nontrivial p-radical subgroups of G. (Recall a p-radical
subgroup of G is a p-subgroup P of G such that P NG (P )=P has no nontrivial
normal p-subgroups.)
G acts on each of these posets by conjugation and thus from each of these G-
posets, one gets a G-simplicial complex Sp(G), Ap(G) and Bp(G) respectively.
Sp(G) is usually called the Brown complex of G and Ap(G) is usually called
the Quillen complex of G where the dependence on the prime p is understood.
Notice again that the trivial subgroup is not included in any of these posets, *
*since
if it were the resulting complex would be a cone and hence trivially contractib*
*le.
It was shown via work of Quillen and Thevenaz, that Sp(G) and Ap(G) are
G-homotopy equivalent and via work of Bouc and Thevenaz that Bp(G) and
Sp(G) are G-homotopy equivalent. Thus, in a sense, these three G-complexes
capture the same information.
Recall the following elementary yet very important lemma (see [B ]):
4
Lemma 2.1. If f0; f1 : P1 ! P2 are two maps of posets such that f0(x) f1(x)
for all x 2 P1 then the simplicial maps induced by f0 and f1 from |P1| to |P2|
are homotopic.
Using this, Quillen made the following observation, if P0 is a nontrivial
normal p-subgroup of G, then we may define a poset map f : sp(G) ! sp(G)
by f(P ) = P0P and by the lemma above, f would be homotopic to the identity
map, but on the other hand since f(P ) contains P0 always, again by the lemma,
f is also homotopic to a constant map. Thus we see that Sp(G) is contractible
in this case. Quillen then conjectured
Conjecture 2.2 (Quillen). If G is a finite group, Sp(G) is contractible if and
only if G has a nontrivial normal p-subgroup.
He proved his conjecture in the case that G is solvable but the general
conjecture remains open. Notice though that if Sp(G) is G-homotopy equivalent
to a point space then this does imply G contains a nontrivial normal p-subgroup
since in this case Sp(G)G is homotopy equivalent to a point which means in
particular Sp(G)G is not empty, yielding a nontrivial normal p-subgroup.
The purpose of this paper is to introduce some simplicial complexes associ-
ated to elements of a group rather than subgroups of a group and use these to
give a different perspective on some of the complexes above.
For this purpose we make the following definitions:
Definition 2.3.Let G be a group. Define a simplicial complex C(G) by declar-
ing a n-simplex in this complex to be a collection [g0; g1; : :;:gn] of distinc*
*t non-
trivial elements of G, which pairwise commute.
Similarly define a simplicial complex NC(G) by declaring a n-simplex to be
a collection [g0; g1; : :;:gn] of nontrivial elements of G, which pairwise do n*
*ot
commute.
It is trivial to verify that the above definition, does indeed define comple*
*xes
on which G acts simplicially by conjugation.
Usually when one studies simplicial group actions, it is nice to have admiss*
*i-
ble actions, i.e., actions where if an element of G fixes a simplex, it actuall*
*y fixes
it pointwise. Although C(G) and NC(G) are not admissible in general, one can
easily fix this by taking a barycentric subdivision. The resulting complex is of
course G-homotopy equivalent to the original, however it now is the realization
of a poset.
Thus if we let P C(G) be the barycentric subdivision of C(G), it corresponds
to the realization of the poset consisting of subsets of nontrivial, pairwise c*
*om-
muting elements of G, ordered by inclusion. Similarly if we let P NC(G) be
the barycentric subdivision of NC(G), it corresponds to the realization of the
poset consisting of subsets of nontrivial, pairwise non-commuting elements of
G, ordered by inclusion.
Depending on the situation, one uses either the barycentric subdivision or
the original. For purposes of understanding the topology, the original is easier
but for studying the G-action, the subdivision is easier.
5
Of course, we will want to work a prime at a time also, so we introduce the
following p-local versions of C(G) and NC(G).
Definition 2.4.Given a group G and a prime p, let Cp(G) be the subcomplex
of C(G) where the simplices consist of sets of nontrivial, pairwise commuting
elements of order p.
Similarly let NCp(G) be the subcomplex of NC(G) where the simplices con-
sist of sets of nontrivial, pairwise non-commuting elements of order p
Of course the same comments about the G-action and the barycentric sub-
division apply to these p-local versions.
Our first order of business is to see that the commuting complexes C(G) and
Cp(G) are nothing new. We will find the following standard lemma useful for
this purpose (see [B ]):
Lemma 2.5. If f is a G-map between admissible G-simplicial complexes X
and Y with the property that for all subgroups H G, f restricts to an ordinary
homotopy equivalence between XH and Y H (recall XH is the subcomplex of
X which consists of elements fixed pointwise by H), then f is a G-homotopy
equivalence, i.e., there is a G-map g : Y ! X such that f O g and g O f are
G-homotopic to identity maps.
Theorem 2.6. Let G be a finite group, then C(G) is G-homotopy equivalent
to the simplicial realization of the poset A(G) of nontrivial abelian subgroups*
* of
G, ordered by inclusion and acted on by conjugation.
Furthermore if p is a prime, then Cp(G) is G-homotopy equivalent to Ap(G)
(and thus Sp(G) and Bp(G).)
Proof. First we will show homotopy equivalence and remark on G-homotopy
equivalence later.
We work with P C(G), the barycentric subdivision. Notice that the associ-
ated poset of P C(G) contains the poset A(G) of nontrivial abelian subgroups of
G as a subposet, they are merely the commuting sets whose elements actually
form an abelian subgroup (minus identity). Let i : A(G) ! P C(G) denote this
inclusion.
We now define a poset map r : P C(G) ! A(G) as follows: If S is a set
of nontrivial, pairwise commuting elements of G, then < S >, the subgroup
generated by S will be a nontrivial abelian subgroup of G, thus we can set
r(S) = {< S > -1}. It is obvious that r is indeed a poset map, and that
S r(S) and so i O r is homotopic to the identity map of P C(G) by lemma 2.1.
Furthermore it is clear that r O i = Id and so r is a deformation retraction of
P C(G) onto A(G).
Thus P C(G) is homotopy equivalent to A(G). To see this is a G-homotopy
equivalence, we just need to note that r is indeed a G-map, and maps a commut-
ing set invariant under conjugation by a subgroup H into a subgroup invariant
under conjugation by H and thus induces a homotopy equivalence between
P C(G)H and A(G)H for any subgroup H. Thus r is indeed a G-homotopy
equivalence by lemma 2.5.
6
The p-local version follows exactly in the same manner, once one notes that
the subgroup generated by a commuting set of elements of order p is an elemen-_
tary abelian p-subgroup. |__|
Thus we see from theorem 2.6, that the commuting complexes at a prime p
are basically the Ap(G) in disguise. However for the rest of the paper, we look
at the non-commuting complexes and we will see that they are quite different,
and in some sense dual to the commuting ones. However before doing that,
we conclude this section by looking at a few more properties of the commuting
complex.
Recall the following important proposition of Quillen [Q ]:
Proposition 2.7. If f : X ! Y is a map of posets, and y 2 Y we define
f=y= {x 2 X|f(x) y}
y\f= {x 2 X|f(x) y}:
Then if f=y is contractible for all y 2 Y (respectively y\f is contractible for*
* all
y 2 Y ) then f is a homotopy equivalence between |X| and |Y |.
Using this we will prove:
Proposition 2.8. If G is a group with nontrivial center then A(G) and hence
C(G) is contractible. Moreover, A(G) is homotopy equivalent to Nil(G) where
Nil(G) is the poset of nontrivial nilpotent subgroups of G.
Proof. If G has a nontrivial center Z(G), then A(G) is conically contractible
via A AZ(G) Z(G) for any abelian subgroup A of G. Thus C(G) is also
contractible as it is homotopy equivalent to A(G).
Let i : A(G) ! Nil(G) be the natural inclusion of posets. Take N 2 Nil(G)
and let us look at i=N = {B 2 A(G)|B N} = A(N). However since N is
nilpotent, it has a nontrivial center Z and hence A(N) is conically contractibl*
*e.
Thus by proposition 2.7 the result follows. __
|__|
It is natural to ask if:
Conjecture 2.9. C(G) is contractible if and only if G has a nontrivial center.
3 Non-commuting complexes
Fix a group G, let us look at NC(G). The first thing we notice is that any
nontrivial central element in G gives us a point component in NC(G) and hence
is not so interesting. Thus we define:
Definition 3.1.BNC(G) is the subcomplex of NC(G) consisting of those sim-
plices of NC(G) which are made out of noncentral elements. Thus BNC(G) is
empty if G is an abelian group.
7
The first thing we will show is that if G is a nonabelian group, (so that
BNC(G) is nonempty) then BNC(G) is not only path-connected but it is
simply-connected. Notice also that this means the general picture of NC(G) is
as a union of components, with at most one component of positive dimension
and this is BNC(G) and it is simply-connected. Also notice that BNC(G) is
invariant under the conjugation G-action, and the point components of NC(G)
are just fixed by the G-action as they correspond to central elements.
Theorem 3.2. If G is a nonabelian group, then BNC(G) is a simply-connected
G-simplicial complex.
Proof. First we show that BNC(G) is path-connected. Take any two vertices
in BNC(G), call them g0 and g1, then these are two noncentral elements of G,
thus their centralizer groups C(g0) and C(g1) are proper subgroups of G.
It is easy to check that no group is the union of two proper subgroups for
suppose G = H [ K where H; K are proper subgroups of G. Then we can find
h 2 G - K (it follows h 2 H) and k 2 G - H (hence k 2 K). Then hk is not
in H as h 2 H and k =2H. Similarly hk =2K so hk =2H [ K = G which is an
obvious contradiction. Thus no group is the union of two proper subgroups.
Thus we conclude that C(g0) [ C(g1) 6= G and so we can find an element
w which does not commute with either g0 or g1 and so the vertices g0 and g1
are joined by an edge path [g0; w] + [w; g1]. (The + stands for concatenation.)
Thus we see BNC(G) is path-connected. In fact, any two vertices of BNC(G)
can be connected by an edge path involving at most two edges of BNC(G).
To show it is simply-connected, we argue by contradiction. If it was not
simply connected, then there would be a simple edge loop which did not contract,
i.e., did not bound a suitable union of 2-simplices. (a simple edge loop is for*
*med
by edges of the simplex and is of the form L = [e0; e1] + [e1; e2] + . .+.[en-1*
*; en]
where all the ei are distinct except e0 = en.)
Take such a loop L with minimal size n. (Notice n is just the number of
edges involved in the loop.)
Since we are in a simplicial complex, certainly n 3.
Suppose n > 5, then e3 can be connected to e0 by an edge path E involving
at most two edges by our previous comments. This edge path E breaks our
simple edge loop into two edge loops of smaller size which hence must contract
since our loop was minimal. However, then it is clear that our loop contracts
which is a contradiction so n 5.
So we see 3 n 5 so we have three cases to consider:
(a) n = 3:
Here L = [e0; e1] + [e1; e2] + [e2; e3] with e3 = e0. But then it is easy to
see {e0; e1; e2} is a set of pairwise non-commuting elements and so gives us a
2-simplex [e0; e1; e2] in BNC(G) which bounds the loop which gives a contra-
diction.
(b) n = 4:
Here L = [e0; e1] + [e1; e2] + [e2; e3] + [e3; e4] with e4 = e0. Thus L forms a
square. Notice by the simplicity of L, the diagonally opposite vertices in the
8
square must not be joined by an edge in BNC(G), i.e., they must commute,
thus e0 commutes with e2 and e1 commutes with e3.
Since e0 and e1 do not commute, {e0; e1; e0e1} is a set of mutually non-
commuting elements and so forms a 2-simplex of BNC(G). Since e2 commutes
with e0 but not with e1, it does not commute with e0e1 and thus {e0e1; e1; e2}
also is a 2-simplex in BNC(G). Similar arguments show that {e0e1; e0; e3} and
{e0e1; e2; e3} form 2-simplices in BNC(G). The union of the four 2-simplices
mentioned in this paragraph, bound our loop giving us our contradiction. Thus
we are reduced to the final case:
(c) n = 5:
Here L = [e0; e1] + [e1; e2] + [e2; e3] + [e3; e4] + [e4; e5] with e5 = e0. Th*
*us L
forms a pentagon, and again by simplicity of L, nonadjacent vertices in the
pentagon cannot be joined by an edge in BNC(G), thus they must commute.
Similar arguments as for the n = 4 case yield that [e0; e1; e0e1], [e0e1; e1; e*
*2] and
[e0e1; e0; e4] are 2-simplices in BNC(G) and the union of these three simplices
contract our loop L into one of length four namely [e0e1; e2] + [e2; e3] + [e3;*
* e4] +
[e4; e0e1] which by our previous cases, must contract thus yielding the_final
contradiction. |__|
From theorem 3.2, we see that BNC(G) is simply-connected for any non-
abelian group G. One might ask if it is contractible? The answer is no in gener*
*al
although there are groups G where it is contractible. We look at these things
next:
Proposition 3.3. For a general finite nonabelian group G, the center Z(G) of
G acts freely on BNC(G) by left multiplication and hence |Z(G)| divides the
Euler characteristic of BNC(G).
For a group of odd order, the simplicial map A which maps a vertex g to
g-1 is a fixed point free map on BNC(G) and on NC(G) of order 2. Thus the
Euler characteristic of both BNC(G) and NC(G) is even in this case.
Thus if BNC(G) is F-acyclic for some field F, then G must have trivial
center and be of even order.
Proof. The remarks about Euler characteristics follow from the fact that if a
finite group H acts freely on a space where the Euler characteristic is defined,
then |H| must divide the Euler characteristic. So we will concentrate mainly on
finding such actions.
First for the action of Z(G). a 2 Z(G) acts by taking a simplex [g0; : :;:gn]
to a simplex [ag0; : :;:agn]. Notice this is well-defined since agi is noncent*
*ral
if gi is noncentral and since agi commutes with agj if and only if gi commutes
with gj.
Furthermore if a is not the identity element, this does not fix any simplex
[g0; : :;:gn] since if the set {g0; : :;:gn} equals the set {ag0; : :;:agn} the*
*n ag0 is
one of the gj's but ag0 commutes with g0 so it would have to be g0 but ag0 = g0
gives a = 1, a contradiction.
Thus this action of nonidentity central a does not fix any simplex and so we
get a free action of Z(G) on BNC(G).
9
Now for the action of A, first note that A is well-defined since if [g0; : :*
*;:gn]
is a set of mutually non-commuting elements of G, so is [g-10; : :;:g-1n]. Clea*
*rly
A O A = Id. Furthermore, if the two sets above are equal, then g-10would have
to be one of the gj. But since g-10commutes with g0 it would have to be g0, i.e*
*.,
g0 would have to have order 2. Similarly, all the gi would have to have order
2. Thus in a group of odd order, A would not fix any simplex of BNC(G) or
NC(G), and hence would not have any fixed points.
The final comment is to recall that if BNC(G) were F-acyclic, its Euler
characteristic would be 1 and hence the center of G would be trivial and G__
would have to have even order by the facts we have shown above. |__|
Proposition 3.3 shows that BNC(G) is not contractible if G is of odd order
or if G has a nontrivial center, for example if G were nilpotent. There is a
corresponding p-local version which we state next:
Proposition 3.4. If 2||Z(G)| or if G has odd order then NC2(G) is not con-
tractible, in fact the Euler characteristic is even. NCp(G) is never contractib*
*le
for any odd prime p, in fact it always has even Euler characteristic.
Proof. Follows from the proof of proposition 3.3, once we note that left multip*
*li-
cation by a central element of order 2 takes the subcomplex NC2(G) of NC(G)
to itself and that the map A maps NCp(G) into itself, as the inverse of an ele-
ment has the same order as the element. For odd primes p, A is fixed point_free_
on NCp(G) always as no elements of order 2 are involved in NCp(G). |__|
The fact that BNC(G) can be contractible sometimes is seen in the next
proposition.
Proposition 3.5. If G is a nonabelian group with a self-centralizing involu-
tion i.e., an element x of order 2 such that C(x) = {1; x}, then BNC(G) is
contractible. In fact, BNC(G) = NC(G) in this case.
Thus for example, NC(3) = BNC(3) is contractible where 3 is the
symmetric group on 3 letters.
Proof. Since x does not commute with any nontrivial element, the center of G
is trivial and NC(G) = BNC(G). Furthermore, it is clear that BNC(G) is a __
cone with x as vertex and hence is contractible. |__|
To help show NC(G) of a group is not contractible, we note the following
observation which uses Smith theory. (See [B ]).
Proposition 3.6. If G is a group and NC(G) is F2-acyclic where F2 is the field
with two elements, then NC2(G) is also F2-acyclic. Furthermore one always has
O(NC(G)) = O(NC2(G)) mod 2.
Proof. We first recall that the map A from proposition 3.3 has order 2 as a
map of NC(G). However it might have fixed points, in fact from the proof of
that proposition, we see that A fixes a simplex [g0; : :;:gn] of NC(G) if and
only if each element gi has order 2 and it fixes the simplex pointwise. Thus
10
the fixed point set of A on NC(G) is nothing other than NC2(G) the 2-local
non-commuting complex for G. Since A has order 2, we can apply Smith Theory
to finish the proof of the first statement of the proposition. The identity on *
*the
Euler characteristics follows once we note that under the action of A, the cells
of NC(G) break up into free orbits and cells which are fixed by A and the_fixed
cells exactly form NC2(G). |__|
We observed earlier that if BNC(G) is contractible then the center of G is
trivial, i.e. BNC(G) = NC(G), hence, by proposition 3.6, NC2(G) is F2-acyclic
(and in particular nonempty).
Definition 3.7.Let G be a finite group. We define nc(G) to be the maximum
size of a set of pairwise non-commuting elements in G. Thus nc(G) - 1 is the
dimension of NC(G).
We now compute a general class of examples, the Frobenius groups. Recall a
group G is a Frobenius group if it has a proper nontrivial subgroup H with the
property that H \ Hg = 1 if g 2 G - H. H is called the Frobenius complement
of G. Frobenius showed the existence of a normal subgroup K such that K =
G - [g2G(Hg - {1}). Thus G is a split extension of K by H, i.e., G = K xOEH,
for some homomorphism OE : H ! Aut(K). This K is called the Frobenius
kernel of G.
We have:
Proposition 3.8. If G is a Frobenius group with Frobenius kernel K and Frobe-
nius complement H, then
NC(G) = NC(K) * NC(H)|K|
where * stands for simplicial join and the superscript |K| means that NC(K)
is joined repeatedly with |K| many copies of NC(H).
It also follows that nc(G) = nc(K) + |K|nc(H).
Finally if both H and K are abelian, then NC(G) is homotopy equivalent to
a wedge of (|K| - 2)(|H| - 2)|K|many |K|-spheres.
Proof. First, from the condition that H \ Hg = 1 for g 2 G - H we see that
no nonidentity element of H commutes with anything outside of H. Thus
conjugating the picture, no nonidentity element of any Hg commutes with any-
thing outside Hg. Thus if Hg1; Hg2; : :;:Hgm is a complete list of the con-
jugates of H in G, we see easily that G - {1} is partitioned into the sets
K - {1}; Hg1 - {1}; : :;:Hgm - 1 and two elements picked from different sets
in this partition will not commute. Thus it follows that the non-commuting
complex based on the elements of G - {1} decomposes as a join of the non-
commuting complexes based on each set in the partition. To complete the
picture one notices that each Hg is isomorphic to H and so contributes the
same non-commuting complex as H and furthermore since the conjugates of H
make up G - K, a simple count gives that m = |G|-|K|_|H|-1= |H||K|-|K|_|H|-1= |*
*K|.
11
The sentence about nc(G) follows from the fact that if we define d(S) =
dim (S) + 1 for a simplicial complex S, then d(S1 * S2) = d(S1) + d(S2). Thus
since d(NC(G)) = nc(G) this proves the stated formula concering nc(G).
Finally when H and K are abelian, NC(H) and NC(K) are just a set of
points namely the nonidentity elements in each group. The short exact sequence
of the join together with the fact that NC(G) = BNC(G) is simply-connected_
finishes the proof. |__|
Example 3.9. A4 is a Frobenius group with kernel Z=2Z x Z=2Z and comple-
ment Z=3Z. Thus proposition 3.8 shows NC(A4) ' S4 _ S4 and nc(A4) = 5.
Claim 3.10. NC2(A5) is a 4-spherical complex homotopy equivalent to a bou-
quet of 32 4-spheres. Thus it is 3-connected and has odd Euler characteristic.
Hence NC(A5) has odd Euler characteristic.
Proof. The order of A5 is 60 = 4 . 3 . 5 of course. It is easy to check that th*
*ere
are five Sylow 2-subgroups P which are elementary abelian of rank 2 and self-
centralizing, i.e., CG (P ) = P , and are "disjoint", i.e., any two Sylow subgr*
*oups
intersect only at the identity element.
Thus the picture for the vertices of NC2(A5) is as 5 sets {Si}5i=1of size 3.
(Since each Sylow 2-group gives 3 involutions.) Now since the Sylow 2-groups
are self-centralizing, this means that two involutions in two different Sylow 2-
subgroups, do not commute and thus are joined by an edge in NC2(A5). Thus
we see easily that NC2(A5) is the join S1 * S2 * S3 * S4 * S5.
Using the short exact sequence for the join (see page 373, Exercise 3 in
[M ]), one calculates easily that NC2(A5) has the homology of a bouquet of 32
4-spheres. Since the join of two path connected spaces S1* S2 and S3* S4* S5 is
simply-connected, it follows that NC2(A5) is homotopy equivalent to a bouquet
of 32 4-spheres, and since it is obviously 4-dimensional, this completes all but
the last sentence of the claim. The final sentence follows from proposition 3.6
which says that NC(A5) has the same Euler characteristic as NC2(A5) mod_
2. |__|
At this stage, we would like to make a conjecture:
Conjecture 3.11. If G is a nonabelian simple group, then NC(G) = BNC(G)
has odd Euler characteristic.
Recall the following famous theorem of Feit and Thompson:
Theorem 3.12 (Odd order theorem). Every group of odd order is solvable.
Notice, that if the conjecture is true, it would imply the odd order theorem.
This is because it is easy to see that a minimal counterexample G to the odd
order theorem would have to be an odd order nonabelian simple group. The
conjecture would then say BNC(G) has an odd Euler characteristic and propo-
sition 3.3 would say G was even order which is a contradiction to the original
assumption that G has odd order.
12
4 General commuting structures
Before we further analyze the NC(G) complexes introduced in the last section,
we need to extend our considerations to general commuting structures.
Definition 4.1.A commuting structure is a set S together with a reflexive,
symmetric relation ~ on S. If x; y 2 S with x ~ y we say x and y commute.
Definition 4.2.Given a commuting structure (S; ~), the dual commuting struc-
ture (S; ~0) is defined by
(a) ~0 is reflexive and
(b)For x 6= y, x ~0y if and only if x does not commute with y in (S; ~).
When it is understood we write a commuting structure as S and its dual as
S0. It is easy to see that S00= S in general.
Definition 4.3.Given a commuting structure S, C(S) is the simplicial complex
whose vertices are the elements of S and such that [s0; : :;:sn] is a face of C*
*(S) if
and only if {s0; : :;:sn} is a commuting set i.e., si~ sj for all i and j. Simi*
*larly
we define NC(S) = C(S0) and refer to the elements in a face of NC(S) as a
non-commuting set in S.
Below are some examples of commuting structures which will be important
in our considerations:
(a) The nontrivial elements of a group G form a commuting structure which we
denote also by G, where x ~ y if and only if x and y commute in the group
G. In this case C(G) and NC(G) are the complexes considered in the previous
sections.
(b) The noncentral elements of a group G form a commuting structure G-Z(G)
and NC(G - Z(G)) = BNC(G). Similarly the elements of order p in G form a
commuting structure Gp and C(Gp) = Cp(G), NC(Gp) = NCp(G).
(c) If V is a vector space equipped with a bilinear map [.; .] : V V ! V .
Then the nonzero elements of V form a commuting structure also denoted by
V , where v1 ~ v2 if [v1; v2] = 0 or if v1 = v2.
(d) In the situation in (c), we can also look at the set P (V ) of lines in V .*
* The
commuting structure on V descends to give a well-defined commuting structure
on P (V ), which we will call the projective commuting structure.
(e) If 1 ! C ! G ! Q ! 1 is a central extension of groups with Q abelian
then one can form [.; .] : Q ! C by
[x; y] = ^x^y^x-1^y-1
where x; y 2 Q and ^xis a lift of x in G etc. It is easy to see this is well-de*
*fined,
independent of the choice of lift and furthermore that the bracket [.; .] is bi*
*linear.
13
Thus by (c), we get a commuting structure on the nontrivial elements of Q via
this bracket. We denote this commuting structure by (G; C). Notice in general
it is not the same as the commuting structure of the group G=C which is abelian
in this case. More generally even if Q is not abelian, one can define a commuti*
*ng
structure on the nontrivial elements of Q from the extension by declaring x ~ y
if and only if ^xand ^ycommute in G, we will denote this commuting structure
by (G; C) in general.
Example 4.4. If P is an extraspecial p-group of order p3 then it has center
Z of order p and P=Z is an elementary abelian p-group of rank 2. Let Symp
denote the commuting structure obtained from a vector space of dimension two
over Fp equipped with the symplectic alternating inner product [x; y] = 1 where
{x; y} is a suitable basis. Then it is easy to check that (P ; Z) = Symp.
One of the main results we will use in order to study the non-commuting
complexes associated to commuting structures is a result of A. Bj"orner, M.
Wachs and V. Welker on "blowup" complexes which we describe next:
Let S be a finite simplicial complex with vertex set [n] (This means the
vertices have been labelled 1, : :,:n). To each vertex 1 i n, we assign a
positive integer mi. Let m = (m1; : :;:mn), then the "blowup" complex Sm is
defined as follows:
The vertices of Sm are of the form (i; j) where 1 i n; 1 j mi. (one
should picture mi vertices in Sm over vertex i in S.)
The faces of Sm are exactly of the form [(i0; j0); : :;:(in; jn)] where [i0;*
* : :;:in]
is a face in S. (In particular il6= ik for l 6= k.)
The result of Bj"orner, Wachs and Welker describes Sm up to homotopy
equivalence, in terms of S and its links. More precisely we have:
Theorem 4.5 (Bj"orner, Wachs, Welker [BWW ]). For any connected sim-
plicial complex S with vertex set [n] and given n-tuple of positive integers m =
(m1; : :;:mn) we have:
_
Sm ' S _ [Susp|F|(Lk(F ))]fl(F):
F2S
Here the _ stands for wedge of spaces, Suspk for k-fold suspension, Lk(F ) for
the link of the face F in S, |F | for the number ofQvertices in the face F (whi*
*ch
is one more than the dimension of F ) and fl(F ) = i2F(mi- 1). Thus in the
decomposition above, fl(F ) copies of Susp|F|(Lk(F )) appear wedged together.
Note that in [BWW ], the empty face is considered a face in any complex. In
the above formulation we are not considering the empty face as a face and thus
have separated out the S term in the wedge decomposition.
For example, if we take a 1-simplex [1; 2] as our complex S and use m = (2; *
*2),
then it is easy to see that Sm is a circle. On the other hand all links in S are
contractible except the link of the maximal face [1; 2] which is empty (a "(-1)-
sphere"). Thus everything in the right hand side of the formula is contractible
14
except the 2-fold suspension of this (-1)-sphere which gives a 1-sphere or circ*
*le
as expected.
Also note that whenever some mi = 1, the corresponding fl(F ) = 0 and so
that term drops our of the wedge decomposition. Thus if m1 = . .=.mn = 1,
the decomposition gives us nothing as Sm = S.
Now for some examples of where this theorem applies:
Corollary 4.6. If G is a finite group and Z(G) is its center, then BNC(G) is
the blowup of (G; Z(G)) where each mi= |Z(G)|. This is because everything in
the same coset of the Z(G) in G commutes with each other and whether or not
two elements from different cosets commute is decided in (G; Z(G)). Thus we
conclude
_ |F|
BNC(G) ' NC(G; Z(G)) _ [Susp|F|(Lk(F ))](|Z(G)|-1):
F2NC(G;Z(G))
Before we do say more, let us look at another example of theorem 4.5 in our
context. The proof is the same as that of example 4.6 and is left to the reader.
Corollary 4.7. Let (V; [.; .]) be a vector space over a finite field k equipped*
* with
a nondegenerate bilinear form. Then NC(V ) is a blowup of NC(P (V )) and we
have
_ |F|
NC(V ) ' NC(P (V )) _ [Susp|F|(Lk(F ))](|k|-2):
F2NC(P(V ))
For example, it is easy to see that in Symp, if [x; y] = 0 then x is a scalar
multiple of y. Thus in P (Symp) two distinct elements do not commute and
hence NC(P (Symp)) is a simplex. Thus all of its links are contractible except
the link of the top face which is empty. This top face has (p2-1)=(p-1) = p+1
vertices. Thus following example 4.7, we see
NC(Symp) ' _(p-2)(p+1)Sp:
Remark 4.8. If (V; [.; .]) is a vector space over a finite field equipped wit*
*h a
nondegenerate symmetric inner product corresponding to a quadratic form Q,
then one can restrict the commuting structure induced on P (V ) to the subset S
consisting of singular points of Q, i.e., lines < x > with Q(x) = 0. (S will be
nonempty only if Q is of hyperbolic type.) Among the non-commuting sets in S
are the subsets called ovoids defined by the property that every maximal singul*
*ar
subspace of V contains exactly one element of the ovoid. (See [G ]) Thus these
ovoids are a special subcollection of the facets of NC(S).
In general, we will find the following notion useful:
Definition 4.9.If (S; ~) is a commuting set and x 2 S, we define the central-
izer of x to be
C(x) = {y 2 S|y ~ x}:
15
This allows us to define
Definition 4.10. If (S; ~) is a commuting set, we define an equivalence relation
on the elements of S by x y if C(x) = C(y). The equivalence classes are
called the centralizer classes of (S; ~). We then define the core of S to be the
commuting set (S ; ~) where the elements are the centralizer classes of S and t*
*he
classes [x] and [y] commute in S if and only if the representatives x, y commute
in S.
It is easy to see that NC(S) is the blowup of NC(S ), where for the vertex
[x] 2 NC(S ) there are mx vertices above it in NC(S), where mx is the size of
the centralizer class [x]. Thus once again theorem 4.5 gives us:
Theorem 4.11. Let S be a commuting set and S be its core, and suppose
NC(S ) is connected. Then
_
NC(S) ' NC(S ) _ [Susp|F|(LkF )]fl(F)
F2NC(S)
Q
where fl(F ) = [x]F(mx-1). Here again mx is the size of the centralizer class
[x].
Remark 4.12. Note that if NC(S) is connected, NC(S ) will automatically be
connected as it is the image of NC(S) under a continuous map.
With some abuse of notation, we will call NC(S ) the core of NC(S).
Following the notation above, in a finite group G, the equivalence relation
"has the same centralizer group" partitions G into centralizer classes. The cen-
tral elements form one class and the noncentral elements thus inherit a partiti*
*on.
Notice if [x] is the centralizer class containing x, then any other generato*
*r of
the cyclic group < x > is in the same class. Thus [x] has at least OE(n) elemen*
*ts
where n is the order of x, and OE is Euler's totient function. Thus if n > 2, t*
*hen
[x] contains at least two elements. Also notice that every thing in the coset
xZ(G) is also in [x] so if Z(G) 6= 1 we can also conclude [x] contains at least
two elements.
Definition 4.13. A non-commuting set S in G is a nonempty subset S, such
that the elements of S pairwise do not commute. A maximal non-commuting
set S is a non-commuting set which is not properly contained in any other non-
commuting set of G.
In general, not all maximal non-commuting sets of a group G have the same
size.
One obtains the following immediate corollary of theorem 4.11 (Assume
|G| > 2 for the following results):
Corollary 4.14. Let G be a finite nonabelian group, and let Ss denote the set
of maximal non-commuting sets in G of size s. Then, for s > 1,
X Y 1
rk(Hs-1(BNC(G))) [ (1 - ___)]:
F2Ss x2F mx
16
where mx is the size of the centralizer class containing x.
In particular, if G is an odd order group or if G has a nontrivial center, t*
*hen
H"s-1(NC(G)) 6= 0
whenever G has a maximal non-commuting set of size s.
Proof. Let X be the non-commuting complex associated to the core of BNC(G).
Let us define a facet to be a face of a simplicial complex which is not contain*
*ed
in any bigger faces. Thus the facets of BNC(G) consist exactly of maximal
non-commuting sets of noncentral elements in G.
The first thing to notice is to every facet F of BNC(G), there corresponds
a facet F of X, and further more this correspondence preserves the dimension
of the facet (or equivalently the number of vertices in the facet).
Since the link of a facet is always empty (a (-1)-sphere), in the wedge
decomposition of theorem 4.11, we get the suspension of this empty link as a
contribution. If the facet F has n vertices in it, then we suspend n times to g*
*et
a (n - 1)-sphere. Thus to every maximal non-commuting set of size s, we get
a s - 1 sphere contribution from the corresponding facet in X. In fact we get
fl(F )-many such spheresQfrom the facet F . However above each facet F in X,
there correspond [x]2F(mx) many facets in BNC(G). Thus in the sumQover
the facets of BNC(G) stated in the theorem, we divide fl(F ) by x2F(mx) in
order to count the contribution from the facet F in X the correct number of
times. Q
Notice that __fl(F)_Qx2F(mx)= x2F(1 - _1_mx) _1_2|F|if all the centralize*
*r classes
have size bigger than one. So if G has the property that the size of the centra*
*lizer
classes of noncentral elements is always strictly bigger than one, for example *
*if
G is odd or if G has a nontrivial center, then
2srk(Hs-1(BNC(G))) |Ss|
for all s 2 N. In particular Hs-1(BNC(G)) 6= 0 whenever G has a maximal
non-commuting set of size s. Also observe that "H0(NC(G)) 6= 0 whenever G has
maximal non-commuting set of size 1, i.e. a singleton consisting of a nontrivia*
*l_
central element (except for the trivial case when G has order 2.) |__|
It is easy to see from this proof that a p-local version of corollary 4.14 i*
*s also
true. We state here only the last part of this result for odd primes:
Corollary 4.15. Let p be an odd prime. Then,
"Hs-1(NCp(G)) 6= 0
whenever G has a maximal non-commuting p-set of size s.
The same is true for p = 2 under the additional condition 2||Z(G)|.
17
Remark 4.16. Notice that the conclusion of corollary 4.14 is consistent with
the simple connectedness of BNC(G), because there is no maximal non-commuting
set of size 2. To see this, observe that whenever there is a non-commuting set
{a; b} with two elements, we can form a bigger non-commuting set {a; b; ab}.
Also notice that this is no longer the case for the p-local case. For exampl*
*e,
when G = D8 = , the complex
BNC2(G) is a rectangle with vertices a; b; ac; bc which is the inflated complex
corresponding to maximal 2-set {a; b}. In particular, BNC2(G) is not simply
connected in general.
Remark 4.17. One of the things that corollary 4.14 says is that if one wants
to calculate nc(G), the answer which is obviously the dimension of BNC(G)
plus one can also be determined by finding the highest nonvanishing homology
of BNC(G) in the case when Z(G) 6= 1 or G is odd order. Thus the answer
is determined already by the homotopy type of BNC(G) in this situation. If
Z(G) = 1 and G has even order, this is no longer true, for example NC(3) is
contractible and so does not have any positive dimensional homology.
Sometimes one can show the non-commuting complex for the core of (S; ~)
is contractible as the next lemma shows:
Lemma 4.18. If (S; ~) has a centralizer class [x] where [x] = C(x) then if the
core is S, then NC(S ) is contractible.
Proof. This is because it is easily seen that NC(S ) is a cone on the vertex_[x]
as everything outside [x] does not commute with x as [x] = C(x). |__|
Thus for example:
Example 4.19. Let p be a prime, then the core of BNC(p) is contractible.
Proof. The cycle x = (1; 2; : :;:p) in p has C(x) =< x > by a simple calcula-_
tion. Thus C(x) = [x] and so the result follows from lemma 4.18. |__|
We now study an important special case:
Definition 4.20. A T C-group G is a group where the commuting relation on
the noncentral elements is transitive. This is equivalent to the condition that*
* all
proper centralizer subgroups C(x) G are abelian.
For example any minimal nonabelian group like S3, A4 or an extraspecial
group of order p3.
Corollary 4.21. If (S; ~) is a commuting set where ~ is also transitive (i.e.,
~ is an equivalence relation), then NC(S) is homotopy equivalent to a wedge of
spheres of the same dimension. (We allow the "empty" wedge, i.e., we allow the
case NC(S) is homotopy equivalent to a point). The dimension of the spheres
is equal to n - 1 where n is theQnumber of equivalence classes in (S; ~) and the
number of spheres appearing is ni=1(mi-1) where mi is the size of equivalence
class i.
18
Thus if G is a T C-group, then BNC(G) is homotopy equivalent to a wedge
of spheres of dimension nc(G) - 1 and the number of spheres is given by a
product as above where the mi are the orders of the distinct proper centralizer
groups of G.
Proof. If ~ is an equivalence relation, it is easy to see that the centralizer *
*classes
are exactly the ~ equivalence classes. Thus one sees that the non-commuting
complex for the core, NC(S ), is a simplex. Thus in Theorem 4.11, all terms
drop out except that corresponding to the maximum face in NC(S ) where the
link is empty. This link is suspended to give a sphere of dimension equal to
the number of equivalence classes minus one. The numberQof these spheres
appearing in the wedge decomposition is the product ni=1(mi- 1) where mi
is the size of equivalence class i and the product is over all equivalence clas*
*ses.
(Thus this can be zero if one of the equivalence classes has size one, in which
case the complex is contractible).
In the case of BNC(G), for G a T C-group, one just has to note that C(x) __
is the centralizer class [x] for any noncentral element x. |_*
*_|
Remark 4.22. In Corollary 4.21, one did not actually have to use the general
result of Bjorner, Wachs and Welker since it is easy to see that in this situat*
*ion,
NC(S) is the join of each equivalence class as discrete sets and a simple count
gives the result.
Now notice in the case that (S; ~) has ~ transitive, the above analysis shows
that NC(S) is a join of discrete sets. Thus it is easy to see that NC(S) is
shellable (This is because the facets of NC(S) are just sets where we have
chosen exactly one element from each of the ~ equivalence classes. We can
linearly order the equivalence classes and then lexicographically order the fac*
*ets.
It is easy to check that this is indeed a shelling.)
Given a shelling of a simplicial complex, there are many combinatorial equal-
ities and inequalities which follow (See [S].) Since these are not so deep in t*
*he
above general context, we will point out only the interpretation when applied to
BNC(G). Recall pure shellable just means shellable where all the facets have
the same dimension.
Proposition 4.23. If G is a nonabelian group such that BNC(G) is pure
shellable, e.g., G a T C-group like 3 or A4, then if one sets nci to be the
number of non-commutative sets of non-central elements which have size i, one
has: j
X
(-1)jC(nc(G); j) + (-1)j-kC(nc(G) - k; j - k)nck 0
k=1
for all 1 j nc(G) where C(n; k) is the usual binomial coefficient.
Proof. Follows from a direct interpretation of the inequalities in [S], page 4,
Theorem 2.9. One warning about the notation in that paper is that |oe| means
the number of vertices in oe and the empty face is considered a simplex_in any
complex. |__|
19
Using that nc1 = |G| - |Z(G)|, nc2 = |G|_2(|G| - m) where m is the number
of conjugacy classes in G, one gets for example from the inequality with j = 2
above:
Corollary 4.24. Let G be a nonabelian group with a transitive commuting re-
lation, i.e., if [g; h] = [h; k] = 1, then [g; h] = 1 for every noncentral g; h*
*; k 2 G.
Then,
nc(G)(nc(G) - 1) + |G|(|G| - m) - 2(nc(G) - 1)(|G| - |Z(G)|) 0:
where m denotes the number of conjugacy classes in G.
5 Duality
Let (S; ~) be a finite set with a commuting relation. Suppose the commuting
complex C(S) breaks up as a disjoint union of path components C(S1); : :;:C(Sn)
where of course we are using Si to stand for the vertex set of component i.
Then notice in the corresponding non-commuting complex, NC(S) we have
NC(S) = NC(S1)*. .*.NC(Sn) where * stands for the join operation as usual.
We state this simple but useful observation as the next lemma:
Lemma 5.1 (Duality). Let (S; ~) be a commuting set then if
Gn
C(S) = C(Si)
i=1
F
where stands for disjoint union, then we have
NC(S) = *ni=1NC(Si)
where * stands for join.
Thus in some sense "the less connected C(S) is, the more connected NC(S)
is."
We can apply this simple observation to say something about the complexes
NCp(G) in general.
Theorem 5.2. Let G be a finite group and p a prime such that p||G|. Let P be
a Sylow p-group of G and define N to be the subgroup generated by NG (H) as
H runs over all the nontrivial subgroups of P .
Then NCp(G) is (|G : N| - 2)-connected and in fact it is the |G : N|-fold
join of some complex with itself.
Proof. By Quillen [Q ], if S1; : :;:Sn are the components of Ap(G), then under
the G-action, G acts transitively on the components with isotropy group N
under suitable choice of labelling. Thus the components are all simplicially
equivalent and there are |G : N| many of them.
20
However we have seen that Ap(G) is G-homotopy equivalent to Cp(G) and
so we have the same picture for that complex. Thus Cp(G) is the disjoint union
of |G : N| copies of some simplicial complex S. Thus by lemma 5.1, NCp(G) is
the |G : N|-fold join of the dual of S with itself. To finish the proof one jus*
*t_has
to note that a k-fold join of nonempty spaces is always (k - 2)-connected. |_*
*_|
References
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[BWW] A. Bj"orner, M. Wachs, and V. Welker, A Poset Fiber Theorem,
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[G] A. Gunawardena, Primitive Ovoids in O+8(q), J. Combinatorial Theory,
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[J] N. Jacobson, Basic Algebra I, Freeman, San Francisco, 1974.
[M] J.R. Munkres, Elements of Algebraic Topology, Addison-Wesley Pub-
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[P] L. Pyber, The Number of Pairwise Non-Commuting Elements and the
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Dept. of Mathematics
University of Wisconsin-Madison,
Madison, WI 53706, U.S.A.
E-mail address: pakianat@math.wisc.edu
Dept. of Mathematics & Statistics
McMaster University
Hamilton, ON, Canada
L8S 4K1.
E-mail address: yalcine@math.mcmaster.ca
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