Self-dual modules and real vectors for solvable groups
Let \(F\) be a finite field of characteristic \(p\) and let \(G\) be a finite group of \(p'\)-order. Let \(V\) be a finite-dimensional faithful \(FG\)-module. Let \(v\) be any vector in \(V\) and let \(C_G(v)\) denote the subgroup of \(G\) that fixes \(v\). We say that \(v\) is an \(F\)-real vector if \(V\), considered as an \(FC_G(v)\)-module, is isomorphic to its dual module. (This is equivalent to saying that the Brauer character of \(C_G(v)\) obtained from its action on \(V\) is real-valued.) Interest in the existence of such real vectors arises in the theory of \(p\)-blocks of \(p\)-solvable groups, as we shall now explain. Suppose that \(H\) is a finite group and \(B\) is a \(p\)-block of irreducible characters of \(H\) with defect group \(D\). Brauer's \(k(B)\)-problem is to show that the number of irreducible complex characters of \(H\) in \(B\) is at most \(|D|\). To solve Brauer's \(k(B)\)-problem when \(H\) is \(p\)-solvable, it suffices to assume that \(H=NG\), where \(N\) is a minimal normal self-centralizing elementary Abelian \(p\)-subgroup of \(H\) and \(G\) is a \(p'\)-group. We may consider \(N\) to be a faithful module for \(FG\), where \(F\) is the field of order \(p\). By a theorem of \textit{G.~R.~Robinson} [J. Algebra 172, No. 1, 159-166 (1995; Zbl 0832.20006)], there is an affirmative answer to Brauer's problem if \(N\) has an \(F\)-real vector. The authors consider the problem of showing the existence of an \(F\)-real vector when \(G\) is solvable and \(F\) is any finite field. In their Theorem A, they show that, when \(G\) is solvable, \(F\)-real vectors exist unless \(|F|=3\), \(5\), \(7\) or \(13\). Generalizing earlier work of Gluck and Knörr, they also show in their Theorem B that \(F\)-real vectors exist if \(G\) has a normal \(2\)-complement or if the Fitting subgroup of \(G\) is not a \(2\)-group when \(|F|=3\) or \(5\), and not a \(\{2,3\}\)-group in general (part of this theorem is described in a note at the end of the paper, improving on what was proved in the paper). In the preliminary reductions for proving the existence of \(F\)-real vectors (where \(G\) need not be solvable), one reaches a situation where \(G\) has a minimal non-Abelian normal subgroup \(E\) which acts irreducibly on \(V\). \(E\) is either quasimple or extraspecial. Following later work of \textit{U.~Riese} in the quasisimple case [J. Algebra 235, No. 1, 45-65 (2001; Zbl 0977.20002)], and Gluck and Maagard in the extraspecial case, it is now known that Brauer's \(k(B)\)-problem has an affirmative answer for \(p\)-solvable groups unless possibly \(p\) is one of \(3\), \(5\), \(7\), \(11\), \(13\), \(19\) or \(31\).
scientific article; zbMATH DE number 1448641