Quillen complex

Definition
For a [finite group]] $$G$$, let $$A_p(G)$$ be the set of non-identity elementary Abelian $$p$$-subgroups of $$G$$. The Quillen complex is the order complex for $$A_p(G)$$.

More explicitly, it is a simplicial complex whose vertices are the elements of $$A_p(G)$$, and where an $$r$$-simplex is a set of vertices which correspond to an ascending chain of subgroups.

Related complexes

 * Brown complex is a bigger complex containing this, which contains all the $$p$$-subgroups rather than just the elementary Abelian ones.
 * Commuting complex is a closely related complex which contains only subgroups of order $$p$$ with adjacence being defined by element-wise commutation.