ON THE POSET OF CONJUGACY CLASSES OF SUBGROUPS OF PI-POWER INDEX

We investigate the poset S-pi(G)/G of conjugacy classes of subgroups o f pi-power index in a finite group G. In particular, we are concerned with combinatorial and topological: properties of tile order complex o f S pi(G)/G. We show that the order complex of S-pi(G)/G is homotopic to a join of orbit spaces of order complexes of posets, which bear str uctural information on the chief factors of tile group. Moreover, for pi-solvable groups and in can pi = {p} we reveal a shellable subposet of S-pi(G)/G of the same homotopy type. This complements the study of the poset S-pi(G) of subgroups of pi-power index performed in [20]. Fo r the analysis of the order complexes we develop some new lemmata on t he topology of order complexes of posets and in the theory of shellabi lity.