Dual modules and group actions on extra-special groups
The author shows how to construct certain examples of groups with a normal extra-special subgroup. Such examples are useful in character, representation, and solvable group theory. Let \(G\) be a normal subgroup of a group \(H\) of index two. Suppose that \(V\) is an irreducible \(FG\)-module over a field \(F=GF(p)\) for a prime \(p\). Suppose that \(V\) is conjugate to its dual module \(V^ \wedge\) in \(H\). That is, the \(FH\)-module \(W\) induced by \(V\) contains copies of both \(V\) and \(V^ \wedge\). Then there is an extra-special group \(E\) on which \(H\) acts such that (a) \(E/Z(E)\) is isomorphic with \(W\), (b) with this isomorphism, the inverse image in \(E\) of \(V\otimes 1\) in \(E\) is abelian, and (c) \(G\) acts trivially on \(Z(E)\) and \(H\backslash G\) either inverts or is trivial on \(Z(E)\). In this last case, \(E\) may be chosen to satisfy either condition except when an absolutely irreducible constituent of \(V\) is \(H\)-conjugate to its contragredient. Viewing \(F\) as a trivial module, \(V^ \wedge\) acts upon \(V\oplus F\). The semidirect product \((V\oplus F)V^ \wedge\) admits \(G\) and is \(E\). Since \(W\) is isomorphic with \(V\oplus V^ \wedge\) as an \(FG\)-module, and since \(H\) acts on \(E/Z(E)\) as it does on \(W\), the proof consists of an examination of how action of \(H\) might lift from \(E/Z(E)\) to \(E\). Form primitive modules may be divided into three kinds [the reviewer, On the structure of a representation of a finite solvable group (1985; Zbl 0562.20002), Theorem 4.12]. This example sheds light on the second two kinds.
scientific article; zbMATH DE number 3485