Let A and B be two proper subsets of ℤn such that A + B ≠ ℤn. The Cauchy-Davenport Theorem states that |A + B| ≥ |A| + |B| − 1 for a prime n. Chowla obtained a generalization for a composite n. Shatrowsky proved the following generalization of these two results.
Let A and B be two commuting non-void finite subsets of a group G. Then |A ∪ AB| ≥ Min(|A| + |B|, o(x); x ∈ B).
We generalize the Shatrowsky theorem as follows.
Let A and B be any two finite non-void subsets of a group. Then either A ∪ AB = A B or |A ∪ AB| ≥ |A| + Min(|B|, o(x); x ∈ B), where B is the subgroup generated by B. In particular, if B generates G, then either A ∪ AB = G or |A ∪ AB| ≥ |A| + min (|B|, o(x); x ∈ B).