I recently realised (rather late in life) that I don't have a good grasp of the following:
Let $G$ be a finite non-abelian group and let $A, B$ be subsets of $G$. (Neither $A$ nor $B$ is required to be a subgroup.) How small can $|AB|$ be?
By $AB$ here I mean the set $AB = \{ab : a \in A, \, b \in B\}$. I impose the condition that $G$ be non-abelian because if $G$ is abelian then we have an answer to the question. This is known as Kneser's theorem and states that (in multiplicative notation):
Let $G$ be a finite abelian group and let $A, B$ be subsets of $G$. Then $$|AB| \geqslant |A|+|B| - |H(AB)|,$$ where $H(AB) := \{g \in G : g(AB) = AB\}$ is the so-called stabiliser of the subset $AB$.
Is there anything known about this question? For example, does Kneser's theorem fail for non-abelian groups?
Kneser's theorem, as I have stated it (and restricted to the setting of finite groups), is (I think) equivalent to the statement that appears on Wikipedia.
There exist easy (and silly) answers to the question. For instance, one could say that $|AB| \geq |A|, |B|$ and thus $|AB| \geq \max\{|A|, |B|\}$. I am not interested in something like that.