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 {aiGi} k i=1 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 k > m + f([G : ⋂ k i=1 Gi]) holds, where f( ∏ r t=1 pt t ) = ∑ r t=1 αt(pt − 1) 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 ⋂ k i=1 Gi to G whose factors are of prime orders.