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Cauchy-Davenport type inequalities, I

Let $\mathbb G = (G, +)$ be a group (either abelian or not). Given $X, Y \subseteq G$, we denote by $\langle Y \rangle$ the subsemigroup of $\mathbb G$ generated by $Y$, and we set $$γ(Y) := \sup_{y_0 \in Y} \inf_{y_0 \ne y \in Y} {\rm ord}(y - y_0)$$ if $|Y| \ge 2$ and $γ(Y) := |Y|$ otherwise. We prove that if $\langle Y \rangle$ is commutative, $Y$ is non-empty, and $X+2Y \neq X + Y + y$ for some $y \in Y$, then $$ |X+Y| \ge |X|+\min(γ(Y), |Y| - 1). $$ Actually, this is obtained from a more general result, which improves on previous work of the author on sumsets in cancellative semigroups, and yields a comprehensive generalization, and in some cases a considerable strengthening, of various additive theorems, notably including the Chowla-Pillai theorem (on sumsets in finite cyclic groups) and the specialization to abelian groups of the Hamidoune-Shatrowsky theorem.