Noncommutative sets of small doubling

One sees that, as a corollary of Kneser’s theorem, any finite non-empty subset A of an abelian group G = ( G , + ) with | A + A | ≤ ( 2 − ε ) | A | can be covered by at most 2 ε − 1 translates of a finite group H of cardinality at most ( 2 − ε ) | A | . Using some arguments of Hamidoune, we establish an analogue in the noncommutative setting. Namely, if A is a finite non-empty subset of a nonabelian group G = ( G , ⋅ ) such that | A ⋅ A | ≤ ( 2 − ε ) | A | , then A is either contained in a right-coset of a finite group H of cardinality at most 2 ε | A | , or can be covered by at most 2 ε − 1 right-cosets of a finite group H of cardinality at most | A | . We also note some connections with some recent work of Sanders and of Petridis.