On Commuting and Non-Commuting Complexes

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 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.