On the homotopy type of the subgroup complex

The aim of this note is to introduce the latest results of research on the homotopy equivalence of a subgroup complex. Let P be a poset(= partially ordered set). The order complex of P is denoted by the symbol ∆(P ); this is the simplicial complex whose k-dimensional simplices are the non-empty chains x0 < x1 < · · · < xk of P . For a finite group G and a prime number p dividing its order, the Brown complex (respectively, Quillen complex) of G at p is defined as the order complex ∆(Sp(G))(resp. ∆(Ap(G))), where Sp(G) = {non-trivial p-subgroup of G}, (resp. Ap(G) = {non-trivial elementary abelian p-subgroup of G}, )