Union of products of subgroups in finite groups

Let G be a group and A, B, C be subgroups of G. In this paper, we prove that if G=AB∪BC, then G = AB or G = BC. But, in general, G=AB∪AC does not imply that G = AB or G = AC. We also prove that if G is a 2-decomposed group, and G is a Sylow tower group or G is nilpotent-by-nilpotent, then G=AB∪AC implies that G = AB or G = AC. Other interesting special cases are also discussed.