Gap modules for direct product groups

Abstract

Let G be a finite group. A gap G-module V is a finite dimensional real G-representation space satisfying the following two conditions: (1) The following strong gap condition holds: dimVP>2dimVH for all P<H≤G such that P is of prime power order, which is a sufficient condition to define a Gsurgery obstruction group and a G-surgery obstruction. (2)V has only one H-fixed point 0 for all large subgroups H, namely H∊L(G). A finite group G not of prime power order is called a gap group if there exists a gap Gmodule. We discuss the question when the direct product K×L is a gap group for two finite groups K and L. According to [5], if K and K×C2 are gap groups, so is K×L. In this paper, we prove that if K is a gap group, so is K×C2. Using [5], this allows us to show that if a finite group G has a quotient group which is a gap group, then G itself is a gap group. Also, we prove the converse: if K is not a gap group, then K×D2n is not a gap group. To show this we define a condition, called NGC, which is equivalent to the non-existence of gap modules.