Finite covers of groups by cosets or subgroups

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ 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 $G_1,...,G_k$ are normal Hall subgroups of $G$, then $k\geq m+f([G:\bigcap_{i=1}^kG_i])$, where $f(\prod_{t=1}^r p_t^{\alpha_t})=\sum_{t=1}^r\alpha_t(p_t-1)$ if $p_1,...,p_r$ are distinct primes and $\alpha_1,...,\alpha_r$ are nonnegative integers. When all the $a_i$ are the identity element of $G$ and all the $G_i$ are subnormal in $G$, we prove that there is a composition series from $\bigcap_{i=1}^kG_i$ to $G$ whose factors are of prime orders. The paper also includes some other results and two challenging conjectures.