Commuting complex of a finite group

Origin
The notion of commuting complex of a finite group was introduced by Alperin in his paper A Lie approach to finite groups.

Definition
Given a group $$G$$ and a prime number $$p$$ that divides the order of $$G$$, define the commuting complex of $$G$$ with respect to $$p$$ is defined as follows:


 * The vertices (or 0-simplices) are the subgroups of order $$p$$.
 * A subset of $$r$$ subgroups of order $$p$$ form an $$(r-1)$$-simplex if and only if they commute pairwise.

Related complexes

 * Brown complex is of the same homotopy type

Article links

 * MathSciNet link for Alperin's article
 * AustMS link for the Coverings and Coverings paper