On the sum of two sets in a group

Abstract Sums C = A + B of two finite sets in a (generally non-abelian) group are considered. The following two theorems are proved. 1. ∣C∣ ≥ ∣A∣ + 1 2 ∣B∣ unless C + (−B + B) = C; 2. There is a subset S of C and a subgroup H such that ∣S∣ ≥ ∣A∣ + ∣B∣ − ∣H∣, and either H + S = S or S + H = S.