σ-properties of finite groups in polynomial time

Abstract

Let H, K be subgroups of the permutation group G of degree n with K E G and σ be a partition of the set of all different prime divisors of |G/K|. We prove that in polynomial time (in n) one can check G/K for σ-nilpotency and σ-solubility; H/K for σ-subnormality and σ-p-permutability in G/K. Moreover one can find the least partition σ of π(G/K) for which G/K is σ-nilpotent. Also one can find the least partition σ of π(G/K) for which H/K is σ-p-permutable in G/K.