FINITE COVERS OF GROUPS BY COSETS OR SUBGROUPS

Abstract

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let a1G1,…,akGk be left cosets in a group G such that [Formula: see text] covers each element of G at least m times but none of its proper subsystems does. We show that if G is cyclic, or G is finite and G1,…,Gk are normal Hall subgroups of G, then the inequality [Formula: see text] holds, where [Formula: see text] if p1,…,pr are distinct primes and α1,…,αr are nonnegative integers. When all the ai are the identity element of G and all the Gi are subnormal in G, we prove that there is a composition series from [Formula: see text] to G whose factors are of prime orders. The paper also includes some other results and two challenging conjectures.