On Commuting and Non-Commuting Complexes

Jonathan Pakianathan and Ergün Yalçın

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. Björner, 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 H_{n−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 NC_{p}(G) and C_{p}(G). We observe that C_{p}(G) is homotopy equivalent to the Quillen complexes A_{p}(G), and obtain some interesting results for NC_{p}(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 {g_{1}, …, g_{n}} such that g_{i} does not commute with g_{j} for i ≠ 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].)

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. Erdös, L. Pyber [P] has also shown that there is a positive constant c such that

cc(G) ≤ |G : Z(G)| ≤ c^{nc(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 BNC_{2}(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. Björner, 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 ∨ ⋁_{F ∈S}[Susp^{|F |}(LkF)]^{γ(F)}

where the F are the faces of S, Lk stands for the link of a face, Susp^{k} stands for a k-fold suspension and ∨ stands for wedge product. The number γ(F) = ∏_{[x]⊆F} (m_{x} − 1) where m_{x} 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 = Σ_{p}, the symmetric group on p letters, for some prime p.

The following is an important consequence of the above decomposition formula:

Result 3 ( 4.14). Let G be a finite nonabelian group, and let S_{s} denote the set of maximal non-commuting sets in G of size s. Then, for s > 1,

rk(H_{s−1}(BNC(G))) ≥ ∑_{F ∈S_{s}}[ ∏_{x∈F}(1 − \frac{1}{m_{x}})]

where m_{x} 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| ≠ 2), then

H̃_{s−1}(NC(G)) ≠ 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 H̃_{s−1}(NC_{p}(G)) ≠ 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| ≠ 2. Observe that this result has a striking formal similarity (in terms 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

H̃_{s−1}(A_{p}(G)) ≠ 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, C_{p}(G), the commuting complex formed by the elements of prime order p, is easily shown to be G-homotopy equivalent to Quillen’s complex A_{p}(G). However, the definition of C_{p}(G) helps us to see a duality between NC_{p}(G) and C_{p}(G) where NC_{p}(G) is the subcomplex of NC(G) spanned by the vertices which correspond to elements of order p. Using a result of Quillen on A_{p}(G), we obtain:

Result 4 ( 5.2). Let G be a group and p a prime with p||G|. Pick a Sylow psubgroup P of G and define N to be the subgroup generated by the normalizers N_{G}(H) where H runs over all the nontrivial subgroups of P.

Then NC_{p}(G) is (|G : N|−2)-connected. In fact NC_{p}(G) is the |G : N|-fold join of a complex S with itself where S is “dual” to a path-component of A_{p}(G).

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 ∈ 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 p_{0} < p_{1} < ⋯ < p_{n}. This is called the simplicial realization of the poset P.

Furthermore any map of posets f : (P_{1}, ≤_{1}) → (P_{2}, ≤_{2}) (map of posets means x_{1} ≤_{1} x_{2} ⇒ f(x_{1}) ≤_{2} f(x_{2})) yields a simplicial map between |P_{1}| and |P_{2}| and hence one has in general a covariant functor from the category of finite 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, Thévenaz 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 s_{p}(G) of nontrivial p-subgroups of G.

(b) The poset a_{p}(G) of nontrivial elementary abelian p-subgroups of G.

(c) The poset b_{p}(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 N_{G}(P)/P has no nontrivial normal p-subgroups.)

G acts on each of these posets by conjugation and thus from each of these Gposets, one gets a G-simplicial complex S_{p}(G), A_{p}(G) and B_{p}(G) respectively. S_{p}(G) is usually called the Brown complex of G and A_{p}(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 contractible.

It was shown via work of Quillen and Thévenaz, that S_{p}(G) and A_{p}(G) are G-homotopy equivalent and via work of Bouc and Thévenaz that B_{p}(G) and S_{p}(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]):

Lemma 2.1. If f_{0}, f_{1} : P_{1} → P_{2} are two maps of posets such that f_{0}(x) ≤ f_{1}(x) for all x ∈ P_{1} then the simplicial maps induced by f_{0} and f_{1} from |P_{1}| to |P_{2}| are homotopic.

Using this, Quillen made the following observation, if P_{0} is a nontrivial normal p-subgroup of G, then we may define a poset map f : s_{p}(G) → s_{p}(G) by f(P) = P_{0}P and by the lemma above, f would be homotopic to the identity map, but on the other hand since f(P) contains P_{0} always, again by the lemma, f is also homotopic to a constant map. Thus we see that S_{p}(G) is contractible in this case. Quillen then conjectured

Conjecture 2.2 (Quillen). If G is a finite group, S_{p}(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 S_{p}(G) is G-homotopy equivalent to a point space then this does imply G contains a nontrivial normal p-subgroup since in this case S_{p}(G)^{G} is homotopy equivalent to a point which means in particular S_{p}(G)^{G} is not empty, yielding a nontrivial normal p-subgroup.

The purpose of this paper is to introduce some simplicial complexes associated 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 declaring a n-simplex in this complex to be a collection [g_{0}, g_{1}, …, g_{n}] of distinct nontrivial elements of G, which pairwise commute.

Similarly define a simplicial complex NC(G) by declaring a n-simplex to be a collection [g_{0}, g_{1}, …, g_{n}] of nontrivial elements of G, which pairwise do not commute.

It is trivial to verify that the above definition, does indeed define complexes on which G acts simplicially by conjugation.

Usually when one studies simplicial group actions, it is nice to have admissible actions, i.e., actions where if an element of G fixes a simplex, it actually 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 commuting 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.

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 C_{p}(G) be the subcomplex of C(G) where the simplices consist of sets of nontrivial, pairwise commuting elements of order p.

Similarly let NC_{p}(G) be the subcomplex of NC(G) where the simplices consist of sets of nontrivial, pairwise non-commuting elements of order p

Of course the same comments about the G-action and the barycentric subdivision apply to these p-local versions.

Our first order of business is to see that the commuting complexes C(G) and C_{p}(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 X^{H} and Y^{H} (recall X^{H} 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 ∘ g and g ∘ 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 C_{p}(G) is G-homotopy equivalent to A_{p}(G) (and thus S_{p}(G) and B_{p}(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 associated 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 ∘ r is homotopic to the identity map of P C(G) by lemma 2.1. Furthermore it is clear that r ∘ 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 commuting 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.

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 elementary abelian p-subgroup.

Thus we see from theorem 2.6, that the commuting complexes at a prime p are basically the A_{p}(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 ∈ Y we define

\begin{aligned}f/y & = \{x ∈ X|f(x) ≤ y\} \\ y\f & = \{x ∈ X|f(x) ≥ y\}. \end{aligned}

Then if f/y is contractible for all y ∈ Y (respectively y\f is contractible for all y ∈ 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 ∈ Nil(G) and let us look at i/N = {B ∈ A(G)|B ⊆ N} = A(N). However since N is nilpotent, it has a nontrivial center Z and hence A(N) is conically contractible. 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 simplices of NC(G) which are made out of noncentral elements. Thus BNC(G) is empty if G is an abelian group.

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 g_{0} and g_{1}, then these are two noncentral elements of G, thus their centralizer groups C(g_{0}) and C(g_{1}) 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 ∈ G − K (it follows h ∈ H) and k ∈ G − H (hence k ∈ K). Then hk is not in H as h ∈ H and k ∉ H. Similarly hk ∉ K so hk ∉ H ∪ K = G which is an obvious contradiction. Thus no group is the union of two proper subgroups.

Thus we conclude that C(g_{0}) ∪ C(g_{1}) ≠ G and so we can find an element w which does not commute with either g_{0} or g_{1} and so the vertices g_{0} and g_{1} are joined by an edge path [g_{0}, w] + [w, g_{1}]. (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 formed by edges of the simplex and is of the form L = [e_{0}, e_{1}] + [e_{1}, e_{2}] + ⋯ + [e_{n−1}, e_{n}] where all the e_{i} are distinct except e_{0} = e_{n}.)

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 e_{3} can be connected to e_{0} 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 = [e_{0}, e_{1}] + [e_{1}, e_{2}] + [e_{2}, e_{3}] with e_{3} = e_{0}. But then it is easy to see {e_{0}, e_{1}, e_{2}} is a set of pairwise non-commuting elements and so gives us a 2-simplex [e_{0}, e_{1}, e_{2}] in BNC(G) which bounds the loop which gives a contradiction.

(b) n = 4: Here L = [e_{0}, e_{1}] + [e_{1}, e_{2}] + [e_{2}, e_{3}] + [e_{3}, e_{4}] with e_{4} = e_{0}. Thus L forms a square. Notice by the simplicity of L, the diagonally opposite vertices in the square must not be joined by an edge in BNC(G), i.e., they must commute, thus e_{0} commutes with e_{2} and e_{1} commutes with e_{3}.

Since e_{0} and e_{1} do not commute, {e_{0}, e_{1}, e_{0}e_{1}} is a set of mutually noncommuting elements and so forms a 2-simplex of BNC(G). Since e_{2} commutes with e_{0} but not with e_{1}, it does not commute with e_{0}e_{1} and thus {e_{0}e_{1}, e_{1}, e_{2}} also is a 2-simplex in BNC(G). Similar arguments show that {e_{0}e_{1}, e_{0}, e_{3}} and {e_{0}e_{1}, e_{2}, e_{3}} 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 = [e_{0}, e_{1}] + [e_{1}, e_{2}] + [e_{2}, e_{3}] + [e_{3}, e_{4}] + [e_{4}, e_{5}] with e_{5} = e_{0}. Thus 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 [e_{0}, e_{1}, e_{0}e_{1}], [e_{0}e_{1}, e_{1}, e_{2}] and [e_{0}e_{1}, e_{0}, e_{4}] are 2-simplices in BNC(G) and the union of these three simplices contract our loop L into one of length four namely [e_{0}e_{1}, e_{2}] + [e_{2}, e_{3}] + [e_{3}, e_{4}] + [e_{4}, e_{0}e_{1}] 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 nonabelian group G. One might ask if it is contractible? The answer is no in general 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 ∈ Z(G) acts by taking a simplex [g_{0}, …, g_{n}] to a simplex [ag_{0}, …, ag_{n}]. Notice this is well-defined since ag_{i} is noncentral if g_{i} is noncentral and since ag_{i} commutes with ag_{j} if and only if g_{i} commutes with g_{j}.

Furthermore if a is not the identity element, this does not fix any simplex [g_{0}, …, g_{n}] since if the set {g_{0}, …, g_{n}} equals the set {ag_{0}, …, ag_{n}} then ag_{0} is one of the g_{j}’s but ag_{0} commutes with g_{0} so it would have to be g_{0} but ag_{0} = g_{0} 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).

Now for the action of A, first note that A is well-defined since if [g_{0}, …, g_{n}] is a set of mutually non-commuting elements of G, so is [g_{0}^{−1}, …, g_{n}^{−1}]. Clearly A ∘ A = Id. Furthermore, if the two sets above are equal, then g_{0}^{−1} would have to be one of the g_{j}. But since g_{0}^{−1} commutes with g_{0} it would have to be g_{0}, i.e., g_{0} would have to have order 2. Similarly, all the g_{i} 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 NC_{2}(G) is not contractible, in fact the Euler characteristic is even. NC_{p}(G) is never contractible 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 multiplication by a central element of order 2 takes the subcomplex NC_{2}(G) of NC(G) to itself and that the map A maps NC_{p}(G) into itself, as the inverse of an element has the same order as the element. For odd primes p, A is fixed point free on NC_{p}(G) always as no elements of order 2 are involved in NC_{p}(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 involution 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(Σ_{3}) = BNC(Σ_{3}) is contractible where Σ_{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 𝔽_{2}-acyclic where 𝔽_{2} is the field with two elements, then NC_{2}(G) is also 𝔽_{2}-acyclic. Furthermore one always has χ(NC(G)) = χ(NC_{2}(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 [g_{0}, …, g_{n}] of NC(G) if and only if each element g_{i} has order 2 and it fixes the simplex pointwise. Thus the fixed point set of A on NC(G) is nothing other than NC_{2}(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 NC_{2}(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, NC_{2}(G) is 𝔽_{2}-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 ∩ H^{g} = 1 if g ∈ G − H. H is called the Frobenius complement of G. Frobenius showed the existence of a normal subgroup K such that K = G − ∪_{g∈G}(H^{g} − {1}). Thus G is a split extension of K by H, i.e., G = K ×_{ϕ} H, for some homomorphism ϕ : 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 Frobenius 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 ∩ H^{g} = 1 for g ∈ 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 H^{g} commutes with anything outside H^{g}. Thus if H^{g_{1}}, H^{g_{2}}, …, H^{g_{m}} is a complete list of the conjugates of H in G, we see easily that G − {1} is partitioned into the sets K − {1}, H^{g_{1}} − {1}, …, H^{g_{m}} − 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 noncommuting complexes based on each set in the partition. To complete the picture one notices that each H^{g} 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 = \frac{|G|−|K|}{|H|−1} = \frac{|H||K|−|K|}{|H|−1} = |K|.

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(S_{1} ∗ S_{2}) = d(S_{1}) + d(S_{2}). 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. A_{4} is a Frobenius group with kernel ℤ/2ℤ × ℤ/2ℤ and complement ℤ/3ℤ. Thus proposition 3.8 shows NC(A_{4}) ≃ S^{4} ∨ S^{4} and nc(A_{4}) = 5.

Claim 3.10. NC_{2}(A_{5}) is a 4-spherical complex homotopy equivalent to a bouquet of 32 4-spheres. Thus it is 3-connected and has odd Euler characteristic. Hence NC(A_{5}) has odd Euler characteristic.

Proof. The order of A_{5} is 60 = 4 · 3 · 5 of course. It is easy to check that there are five Sylow 2-subgroups P which are elementary abelian of rank 2 and selfcentralizing, i.e., C_{G}(P) = P, and are “disjoint”, i.e., any two Sylow subgroups intersect only at the identity element.

Thus the picture for the vertices of NC_{2}(A_{5}) is as 5 sets {S_{i}}_{i=1}^{5} of 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 2subgroups, do not commute and thus are joined by an edge in NC_{2}(A_{5}). Thus we see easily that NC_{2}(A_{5}) is the join S_{1} ∗ S_{2} ∗ S_{3} ∗ S_{4} ∗ S_{5}.

Using the short exact sequence for the join (see page 373, Exercise 3 in [M]), one calculates easily that NC_{2}(A_{5}) has the homology of a bouquet of 32 4-spheres. Since the join of two path connected spaces S_{1} ∗ S_{2} and S_{3} ∗ S_{4} ∗ S_{5} is simply-connected, it follows that NC_{2}(A_{5}) 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(A_{5}) has the same Euler characteristic as NC_{2}(A_{5}) 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 proposition 3.3 would say G was even order which is a contradiction to the original assumption that G has odd order.

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 ∈ S with x ∼ y we say x and y commute.

Definition 4.2. Given a commuting structure (S, ∼), the dual commuting structure (S, ∼′) is defined by

(a) ∼′ is reflexive and (b)For x ≠ y, x ∼′ y 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 S′. It is easy to see that S′′ = 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 [s_{0}, …, s_{n}] is a face of C(S) if and only if {s_{0}, …, s_{n}} is a commuting set i.e., s_{i} ∼ s_{j} for all i and j. Similarly we define NC(S) = C(S′) 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 G_{p} and C(G_{p}) = C_{p}(G), NC(G_{p}) = NC_{p}(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 v_{1} ∼ v_{2} if [v_{1}, v_{2}] = 0 or if v_{1} = v_{2}.

(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] = x̂ŷx̂^{−1}ŷ^{−1}

where x, y ∈ Q and x̂ is a lift of x in G etc. It is easy to see this is well-defined, independent of the choice of lift and furthermore that the bracket [·, ·] is bilinear.

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 commuting structure on the nontrivial elements of Q from the extension by declaring x ∼ y if and only if x̂ and ŷ commute 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 p^{3} 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 𝔽_{p} 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. Björner, 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 m_{i}. Let m̄ = (m_{1}, …, m_{n}), then the “blowup” complex Sm̄ is defined as follows:

The vertices of Sm̄ are of the form (i, j) where 1 ≤ i ≤ n, 1 ≤ j ≤ m_{i}. (one should picture m_{i} vertices in Sm̄ over vertex i in S.)

The faces of Sm̄ are exactly of the form [(i_{0}, j_{0}), …, (i_{n}, j_{n})] where [i_{0}, …, i_{n}] is a face in S. (In particular i_{l} ≠ i_{k} for l ≠ k.)

The result of Björner, Wachs and Welker describes Sm̄ up to homotopy equivalence, in terms of S and its links. More precisely we have:

Theorem 4.5 (Björner, Wachs, Welker [BWW]). For any connected simplicial complex S with vertex set [n] and given n-tuple of positive integers m̄ = (m_{1}, …, m_{n}) we have:

Sm̄ ≃ S ∨ ⋁_{F ∈S}[Susp^{|F |}(Lk(F))]^{γ(F)}.

Here the ∨ stands for wedge of spaces, Susp^{k} for k-fold suspension, Lk(F) for the link of the face F in S, |F | for the number of vertices in the face F (which is one more than the dimension of F) and γ(F) = ∏_{i∈F} (m_{i} − 1). Thus in the decomposition above, γ(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 m̄ = (2, 2), then it is easy to see that Sm̄ 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 except the 2-fold suspension of this (−1)-sphere which gives a 1-sphere or circle as expected.

Also note that whenever some m_{i} = 1, the corresponding γ(F) = 0 and so that term drops our of the wedge decomposition. Thus if m_{1} = ⋯ = m_{n} = 1, the decomposition gives us nothing as Sm̄ = 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 m_{i} = |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

BNC(G) ≃ NC(G; Z(G)) ∨ ⋁_{F ∈NC(G;Z(G))}[Susp^{|F |}(Lk(F))]^{(|Z(G)|−1)^{|F |}}.

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

NC(V) ≃ NC(P (V)) ∨ ⋁_{F ∈NC(P (V))}[Susp^{|F |}(Lk(F))]^{(|k|−2)^{|F |}}.

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 (p^{2} −1)/(p−1) = p+1 vertices. Thus following example 4.7, we see

NC(Symp) ≃ ∨_{(p−2)^{(p+1)}}S^{p}.

Remark 4.8. If (V, [·, ·]) is a vector space over a finite field equipped with 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 singular 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 ∈ S, we define the centralizer of x to be

C(x) = {y ∈ S|y ∼ x}.

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 (S̄, ∼) where the elements are the centralizer classes of S and the classes [x] and [y] commute in S̄ if and only if the representatives x, y commute in S.

It is easy to see that NC(S) is the blowup of NC(S̄), where for the vertex [x] ∈ NC(S̄) there are m_{x} vertices above it in NC(S), where m_{x} 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 S̄ be its core, and suppose NC(S̄) is connected. Then

NC(S) ≃ NC(S̄) ∨ ⋁_{F ∈NC(S̄)}[Susp^{|F |}(LkF)]^{γ(F)}

where γ(F) = ∏_{[x]⊆F} (m_{x} −1). Here again m_{x} is the size of the centralizer class [x].

Remark 4.12. Note that if NC(S) is connected, NC(S̄) 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(S̄) 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 central elements form one class and the noncentral elements thus inherit a partition.

Notice if [x] is the centralizer class containing x, then any other generator of the cyclic group < x > is in the same class. Thus [x] has at least ϕ(n) elements where n is the order of x, and ϕ is Euler’s totient function. Thus if n > 2, then [x] contains at least two elements. Also notice that every thing in the coset xZ(G) is also in [x] so if Z(G) ≠ 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 noncommuting 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 S_{s} denote the set of maximal non-commuting sets in G of size s. Then, for s > 1,

rk(H_{s−1}(BNC(G))) ≥ ∑_{F ∈S_{s}}[ ∏_{x∈F}(1 − \frac{1}{m_{x}})].

where m_{x} 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, then

H̃_{s−1}(NC(G)) ≠ 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 contained 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 F̄ 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 get 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 γ(F)-many such spheres from the facet F. However above each facet F̄ in X, there correspond ∏_{[x]∈F̄}(m_{x}) many facets in BNC(G). Thus in the sum over the facets of BNC(G) stated in the theorem, we divide γ(F) by ∏_{x∈F} (m_{x}) in order to count the contribution from the facet F̄ in X the correct number of times.

Notice that \frac{γ(F)}{∏_{x∈F} (m_{x})} = ∏_{x∈F} (1 − \frac{1}{m_{x}}) ≥ \frac{1}{2^{|F |}} if all the centralizer classes have size bigger than one. So if G has the property that the size of the centralizer classes of noncentral elements is always strictly bigger than one, for example if G is odd or if G has a nontrivial center, then

2^{s}rk(H_{s−1}(BNC(G))) ≥ |S_{s}|

for all s ∈ ℕ. In particular H_{s−1}(BNC(G)) ≠ 0 whenever G has a maximal non-commuting set of size s. Also observe that H̃_{0}(NC(G)) ≠ 0 whenever G has maximal non-commuting set of size 1, i.e. a singleton consisting of a nontrivial 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 is 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,

H̃_{s−1}(NC_{p}(G)) ≠ 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)|.

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 example, when G = D_{8} = ⟨a, b|a^{2} = b^{2} = c^{2} = 1, [a, b] = c, c central⟩, the complex BNC_{2}(G) is a rectangle with vertices a, b, ac, bc which is the inflated complex corresponding to maximal 2-set {a, b}. In particular, BNC_{2}(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) ≠ 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(Σ_{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 S̄, then NC(S̄) is contractible.

Proof. This is because it is easily seen that NC(S̄) 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(Σ_{p}) is contractible.

Proof. The cycle x = (1, 2, …, p) in Σ_{p} has C(x) =< x > by a simple calculation. 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 S_{3}, A_{4} or an extraspecial group of order p^{3}.

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 the number of equivalence classes in (S, ∼) and the number of spheres appearing is ∏_{i=1}^{n}(m_{i} −1) where m_{i} is the size of equivalence class i.

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 m_{i} 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(S̄), is a simplex. Thus in Theorem 4.11, all terms drop out except that corresponding to the maximum face in NC(S̄) 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 number of these spheres appearing in the wedge decomposition is the product ∏_{i=1}^{n}(m_{i} − 1) where m_{i} is the size of equivalence class i and the product is over all equivalence classes. (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 situation, 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 facets. It is easy to check that this is indeed a shelling.)

Given a shelling of a simplicial complex, there are many combinatorial equalities and inequalities which follow (See [S].) Since these are not so deep in the 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 Σ_{3} or A_{4}, then if one sets nc_{i} to be the number of non-commutative sets of non-central elements which have size i, one has:

(−1)^{j}C(nc(G), j) + ∑_{k=1}^{j}(−1)^{j−k}C(nc(G) − k, j − k)nc_{k} ≥ 0

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 |σ| means the number of vertices in σ and the empty face is considered a simplex in any complex.

Using that nc_{1} = |G| − |Z(G)|, nc_{2} = \frac{|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 relation, i.e., if [g, h] = [h, k] = 1, then [g, h] = 1 for every noncentral g, h, k ∈ 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(S_{1}), …, C(S_{n}) where of course we are using S_{i} to stand for the vertex set of component i.

Then notice in the corresponding non-commuting complex, NC(S) we have NC(S) = NC(S_{1})∗⋯∗NC(S_{n}) 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

C(S) = ⨆_{i=1}^{n} C(S_{i})

where ⨆ stands for disjoint union, then we have

NC(S) = ∗_{i=1}^{n}NC(S_{i})

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 NC_{p}(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 N_{G}(H) as H runs over all the nontrivial subgroups of P.

Then NC_{p}(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 S_{1}, …, S_{n} are the components of A_{p}(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.

However we have seen that A_{p}(G) is G-homotopy equivalent to C_{p}(G) and so we have the same picture for that complex. Thus C_{p}(G) is the disjoint union of |G : N| copies of some simplicial complex S. Thus by lemma 5.1, NC_{p}(G) is the |G : N|-fold join of the dual of S with itself. To finish the proof one just has to note that a k-fold join of nonempty spaces is always (k − 2)-connected.

References

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[BWW] A. Björner, M. Wachs, and V. Welker, A Poset Fiber Theorem, preprint, 2000.

[G] A. Gunawardena, Primitive Ovoids in O_{8}^{+}(q), J. Combinatorial Theory, Series A 89 (2000), 70-76.

[J] N. Jacobson, Basic Algebra I, Freeman, San Francisco, 1974.

[M] J.R. Munkres, Elements of Algebraic Topology, Addison-Wesley Publishing Company, Menlo Park, Calif., 1984.

[P] L. Pyber, The Number of Pairwise Non-Commuting Elements and the Index of the Centre in a Finite Group, J. London Math. Soc. 35(2) (1987), 287-295.

[S] J. Shareshian, On the Shellability of the Order Complex of the Subgroup Lattice of a Finite Group, to appear, Trans. Amer. Math. Soc.

[Q] D. Quillen, Homotopy Properties of the poset of nontrivial p-subgroups of a group, Adv. in Math. 28(2) (1978), 101-128.

[Y] E. Yalçın, Set Covering and Serre’s Theorem on the Cohomology Algebra of a p-Group, preprint, 2000.

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