Is there an infinite group $G$ such there is **not** any sequence $(A_n)$  of its subsets such that always
$$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$
?

[link](http://math.stackexchange.com/questions/855942/a-sequence-of-subsets-of-an-infinite-group?noredirect=1#comment1767603_855942)