Cardinality of product of subsets of a group

Question

Let $A, B$ be finite subsets of a group $G$ (not necessarily finite). Is it true that $|AB| = |BA|$? More generally, is it true that $|ABC| = |ABC| =\cdots$ any permutation of three elements, if $C$ is another finite subset of $G$?

For the first question, I know this to be true by the product formula when $A$ and $B$ are groups (so we not only have equality but a formula even), but I was wondering if this was true in general.

Answer 1

No. I just chose two sets each consisting of three random elements of $S_4$.

$A = \{(1,4,3,2),(1,4,2),(1,4,3)\}$ and $B = \{(1,4,2,3),(1,3),(1,3,4)\}$

and found that $|AB|=9$ and $|BA|=8$.

Answer 2

In $S_3$, let \begin{align*} A&=\{(1\;2),(1\;2\;3)\}\\[4pt] B&=\{(2\;3),(1\;2\;3)\} \end{align*} Then $AB$ has $4$ elements, while $BA$ has $2$ elements.