The gap hypothesis for finite groups which have an abelian quotient group not of order a power of 2

Abstract

For a finite group G, an $¥mathscr{L}$(G)-free gap G-module V is a finite dimensional real G-representation space satisfying the two conditions: (1) V^L = 0 for any normal subgroup L of G with prime power index. (2) dim V^P > 2 dim V^H for any P < H ≤ G such that P is of prime power order. A finite group G not of prime power order is called a gap group if there is an $¥mathscr{L}$(G)-free gap G-module. We give a necessary and sufficient condition for that G is a gap group for a finite group G satisfying that G/[G,G] is not a 2-group, where [G,G] is the commutator subgroup of G.