Finite solvable groups whose Quillen complex is Cohen–Macaulay

Abstract

We will prove that the p-Quillen complex of a finite solvable group with cyclic derived group is Cohen-Macaulay, if p is an odd prime. If p = 2 we prove a similar conclusion, but there is a discussion to be made.2000 Mathematics Subject Classification. primary 20D30; secondary 06A11, 20D10, 57M07.Key words and phrases. p-Subgroup complex; Homotopy type of posets.(ii) Suppose that Ω 1 (P ) is the central product of T and D. If T = 1 or T is dihedral then ∆(A p (G)) is Cohen-Macaulay. If T is semi-dihedral then there exist two distinct reduced homology groups of ∆(A p (G)) which are non-trivial.Let us now fix some notations. If r is an element of a poset P we denote with P >r the poset {q ∈ P | q > r} and say the P >r is an upper interval. Analogously we define P <r , P ≥r and P ≤r . Moreover

Elementary abelian subgroups in p-groups with a cyclic derived subgroup

“…The question whether A p (P ) is hCM when P is a p-group with a cyclic derived subgroup has already been considered by Matucci [10], but with strong restrictions in the case p = 2.…”

“…Following Matucci's arguments given in Section 5 of [10], the following corollary should remain true if G is taken to be a solvable group containing a Sylow p-subgroup with a cyclic derived subgroup.…”