The regular module problem. I

Abstract:

In the study of induced representations the following problem arises: Let $H = AG$ be a finite solvable group and ${\mathbf {k}}$ a field with $\operatorname {char}{\mathbf {k}}\nmid \; |A|$. Let $V$ be an irreducible, faithful, primitive ${\mathbf {k}}[AG]$-module. Suppose $H$ contains a normal extraspecial $r$-subgroup $R$ with $Z(R) \leq Z(H)$ and that $A$ acts faithfully on $R$. Under what conditions does $A$ have a regular direct summand in $V$? In this paper we consider this question under the hypotheses that $G = MR$, where $1 \ne M$ is normal abelian in $AM$, $A$ is nilpotent, $(|A|,|MR|) = (|M|,|R|) = 1$ , and $R/Z(R)$ is a faithful, irreducible $AM$-module. We show that $A$ has at least three regular direct summands in $V$ unless $|A|$, $\exp (M)$, and $r$ satisfy certain very restrictive conditions.