On H-Bases and H-Decompositions of the Finite Solvable and Alternating Groups

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Let G be a finite group such that every composition factor of G is either cyclic or isomorphic to the alternating group on n letters for some integer n. Then for every positive integer h there is a subset A ⊆ G such that |A| ≤ (2h − 1) |G|1/h and Ah = G. The following generalization for the group G also holds: For every positive integer h and any nonnegative real numbers α1, α2, ..., αh so that α1 + α2 + · · · + αh = 1 there are subsets A1, A2, ..., Ah ⊆ G such that |A1| ≤ |G|α1, |Ai| ≤ 2 |G|αi for 2 ≤ i ≤ h and A1A2 · · · Ah = G. In particular, the above conclusions hold if G is finite group and either G is an alternating group or G is solvable.