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 2ε−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 2ε|A|, or can be covered by at most 2ε−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.