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: $\dim V^P>2\dim V^H$ for all $H\le G$ 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 $H∊\mathcal{L}\left(G\right)$. 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 $K\times L$ is a gap group for two finite groups $K$ and $L$. According to [5], if $K$ and $K\times C_2$ are gap groups, so is $K\times L$. In this paper, we prove that if $K$ is a gap group, so is $K\times C_2$. 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 $K\times D_{2n}$ is not a gap group. To show this we define a condition, called NGC, which is equivalent to the non-existence of gap modules.