Gap modules for direct product groups

Dedicated to Professor Masayoshi Kamata on his 60th birthday

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: \dim V^P>2\dim V^H for all P< H≤ G such that P is of prime power order, which is a sufficient condition to define a G-surgery obstruction group and a G-surgery obstruction.
(2) V has only one H-fixed point 0 for all large subgroups H, namely H∈ \mathscr{L}(G). A finite group G not of prime power order is called a gap group if there exists a gap G-module. 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× D_2n is not a gap group. To show this we define a condition, called NGC, which is equivalent to the non-existence of gap modules.