Commuting complex

Definition
Let $$G$$ be a finite group and $$p$$ a prime. The commuting complex of $$G$$ is defined a sa simplicial complex whose points are the subgroups of $$G$$ of order exactly $$p$$, where two points are adjacent (joined by an edge) if and only if the two subgroups of order $$p$$ commute element-wise.

Relation with other complexes

 * Brown complex: This is the complex of all $$p$$-subgroups ordered by inclusion. it is of the same homotopy type as the commuting complex.
 * Quillen complex

Facts
The 1-skeleton of the commuting complex, (that is, only the graph, comprising the vertices and edges) is very important to the study and classification of finite simple groups, and to the cosntruction of the Fischer sporadic simple groups.