The extraspecial case of the \(k(GV)\) problem (Q2750955)

Let \(F\) be a finite field of order \(r\), where \(r\) is odd, and let \(G\) be a finite group of order relatively prime to \(r\). 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 a real vector if \(V\), considered as an \(FC_G(v)\)-module, is isomorphic to its dual module. Following research of a number of authors, the existence of a real vector in \(V\) is relevant to the solution of a special case of Brauer's \(k(B)\)-problem, concerning the number of irreducible complex characters in a \(p\)-block \(B\) of a finite \(p\)-solvable group. Inductive attempts to prove the existence of a real vector in \(V\) have divided the problem into two cases. In one case, \(G\) contains a normal quasisimple subgroup \(E\) that acts absolutely irreducibly on \(V\). Here, \textit{U. Riese}, [J. Algebra 235, No. 1, 45-65 (2001; Zbl 0977.20002)], has shown that real vectors exist in all cases when \(r\) is a power of a prime greater than \(31\) (his main result is much more precise than this brief statement). In the other case, examined in the paper under review, \(G\) contains a normal subgroup \(E\) of extraspecial type which acts absolutely irreducibly on \(V\) (we say that \(E\) is of extraspecial type if it is extraspecial of odd prime exponent or exponent \(4\), or if it is the central product of an extraspecial \(2\)-group with a cyclic group of order \(4\)).NEWLINENEWLINENEWLINEThe authors' main theorem concerns a more general situation than that described above, and we will attempt to explain it now. Let \(E\) be of extraspecial type and let \(V\) be a faithful absolutely irreducible \(FE\)-module. Let \(Z(E)\) denote the centre of \(E\) and let \(|E:Z(E)|=p^{2n}\), where \(p\) is a prime and \(n\) a positive integer. Let \(\varepsilon=1\) if \(p=2\) and \(E\) contains exactly \(2^{2n}+2^n-1\) involutions and let \(\varepsilon=1\) if \(p=2\) and \(E\) contains exactly \(2^{2n}-2^n-1\) involutions. Let \(\varepsilon=0\) in the other cases. Let \(G\) be the normalizer of \(E\) in the group of all automorphisms of \(V\). Then for any subgroup \(H\) of \(G\) of order relatively prime to \(r\), there exists a vector \(v\) such that \(V\) is self-dual as an \(FC_H(v)\)-module, except possibly in the following cases:NEWLINENEWLINENEWLINE(a) \(n=1\), \(p=2\) and \(r\) is one of \(5\), \(7\) and \(13\);NEWLINENEWLINENEWLINE(b) \(n=1\), \(p=3\) and \(r\) is one of \(7\) and \(13\);NEWLINENEWLINENEWLINE(c) \(n=2\), \(p=2\), \(\varepsilon=-1\) and \(r\) is one of \(3\) and \(7\);NEWLINENEWLINENEWLINE(d) \(n=3\), \(p=2\), \(\varepsilon=-1\) and \(r=7\);NEWLINENEWLINENEWLINE(e) \(n\geq 4\), \(p=2\), \(\varepsilon=-1\) and \(r\) is one of \(3\), \(7\), \(11\) and \(19\).NEWLINENEWLINENEWLINEFurthermore, in the following cases, there exists an \(H\) satisfying the hypotheses above such that there is no \(v\) for which \(V\) is self-dual as an \(FC_H(v)\)-module:NEWLINENEWLINENEWLINE(f) \(n=1\), \(p=2\), \(r\) is one of \(5\), \(7\) and \(13\), and \(\varepsilon\) is respectively \(0\), \(-1\) and \(0\);NEWLINENEWLINENEWLINE(g) \(n=1\), \(p=3\), and \(r\) is one of \(7\) and \(13\);NEWLINENEWLINENEWLINE(h) \(n=2\), \(p=2\), \(\varepsilon=-1\) and \(r\) is one of \(3\) and \(7\).NEWLINENEWLINENEWLINESpecializing to solvable groups, the authors prove the following result. Let \(F\) be a finite field of characteristic \(p\) and let \(G\) be a solvable group of order relatively prime to \(p\). Let \(V\) be a faithful irreducible \(FG\)-module. Then provided \(p>13\), \(V\) contains a real vector for the action of \(G\). This proves Brauer's conjecture for a \(p\)-block of a solvable group if \(p>13\).