Computing the order of a solvable permutation group
If a group G of permutations is given by a set of generators usually the first step for its investigation is the determination of a base and a strong generating system by one of the versions of the Schreier-Sims method [cf. \textit{J. S. Leon}; Math. Comput. 35, 941-974 (1980; Zbl 0444.20001)]. In this article the author describes a new method, which works if G is soluble (it runs into an endless cycle if G is not soluble). The method is not only considerably faster in many cases tested, but it also constructs at the same time a ``polycyclic generating sequence'', i.e. a sequence of generators \(y_ 1,...,y_ t\) of G such that with \(H_ i=<y_ i,...,y_ t>\) one has \(H_ i\triangleleft H_{i-1}\) for \(1<i\leq t\) and \(H_ 1=G\). There are a number of computational problems, e.g. the determination of the conjugacy classes, for which rather efficient solutions are known for soluble groups using such a polycyclic generating sequence, while they are notoriously difficult using only base and strong generating set. Like other computational methods for soluble groups this one exploits the fact that an orbit of a normal subgroup of a group G is a block for G.
scientific article; zbMATH DE number 4148389
scientific article