A generalization of an addition theorem of shatrowsky

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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 |AAB| ≥ Min(|A| + |B|, o(x); xB).

We generalize the Shatrowsky theorem as follows.

Let A and B be any two finite non-void subsets of a group. Then either AAB = A B or |AAB| ≥ |A| + Min(|B|, o(x); xB), where B is the subgroup generated by B. In particular, if B generates G, then either AAB = G or |AAB| ≥ |A| + min (|B|, o(x); xB).

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