On the finite spaces of non-trivial proper subgroups of finite groups

Abstract

In this paper, we investigate the homotopy properties of $L\left(G\right)$, the finite topological space consisting of all non-trivial proper subgroups of a finite group G. For some classes of groups G, we give the relations between the contractibility of $L\left(G\right)$ and the algebraic properties of G, which is inspired by the study of R. E. Stong on $S_p\left(G\right)$ and $A_p\left(G\right)$.