On the extraspecial case of the \(k(GV)\)-conjecture
This is a further contribution by the author to the solution of Brauer's \(k(B)\)-problem for a finite \(p\)-solvable group. An extraspecial \(2\)-group of order \(2^{2n+1}\) is said to be of negative type if it is the central product of \(n-1\) dihedral groups of order \(8\) and one quaternion group of order 8. Such a group has a unique irreducible faithful representation, which is of symplectic type and has degree \(2^n\). In the analysis of minimal configurations relating to an inductive solution to Brauer's \(k(B)\)-problem for a \(p\)-solvable group \(G\), the case where \(G\) contains a minimal non-Abelian normal subgroup \(E\) which is an extraspecial 2-group of negative type presents various difficulties. In particular, the primes \(3\), \(7\), \(11\) and \(19\) prove to be problematic. The author rules out all possible exceptions relating to this configuration, by proving the following theorem. Let \(p\) be an odd prime and let \(G\) be a finite \(p'\)-group. Let \(V\) be a finite-dimensional faithful \(FG\)-module, where \(F\) is the field of order \(p\). Suppose that \(G\) contains a minimal non-Abelian normal subgroup \(E\) which is extraspecial of order \(2^{2n+1}\) and of negative type. Suppose also that \(E\) acts absolutely irreducibly on \(V\). Then \(V\) contains real vectors for \(G\) provided that \(n\geq 4\). Here we recall that a vector \(v\) in \(V\) is said to be real if \(V\), considered as an \(FC_G(v)\)-module, is isomorphic to its dual module (\(C_G(v)\) is the subgroup of \(G\) that fixes \(v\)). The proof of this result requires some quite delicate analysis that relies in part on information contained in the Atlas of Finite Groups. The finite orthogonal groups over the field of two elements command particular attention.
scientific article; zbMATH DE number 1753062