Title: On Commuting and Non-Commuting Complexes

(to appear in Journal of Algebra)

Authors: Jonathan Pakianathan and Erg\"un Yal\c c\i n

2000 Mathematics Subject Classification.
Primary: 20J05; Secondary: 06A09, 05E25.

Addresses:

Department of Mathematics
University of Rochester
N.Y., U.S.A.

Department of Mathematics
Bilkent University
Ankara, Turkey


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.
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.