Normalizers of \(p\)-subgroups in finite groups.
If \(G\) is a permutation group stored in a computer as a set of generating permutations, and \(H\) is a subgroup of \(G\), the standard way to compute \(N_G(H)\) is to compute a base and strong generating set first. This permits use of the powerful methods pioneered by Sims in the late 1960s to compute almost anything in the group. These methods are so efficient that if \(G\) is a matrix group, the usual way to proceed is to first convert \(G\) into a permutation group. However, it often turns out that the resulting permutations are unmanageably huge. Therefore it is desirable to develop other methods for matrix groups (or more generally, black-box groups). The paper under review reports on an implementation of such methods for \(H\) a \(p\)-group. The method for \(p=2\) is part of the folklore [see for example \textit{G. Butler}, J. Algebra 69, 67-81 (1981; Zbl 0457.20024)], but is here presented more systematically, and completely automated. Essentially, a number of composition series for \(H\) are computed, and for each one a set of random elements are found which centralize each composition factor. Since each composition factor is \(C_2\), this is essentially an involution centralizer calculation. For \(p\) odd, the same strategy is employed, but using the authors' method [described in Arch. Math. 85, No. 6, 485-489 (2005; Zbl 1137.20301)] for finding centralizers of odd-order elements.
scientific article; zbMATH DE number 5551217
scientific article