Let, $G$ be a finite group with $A\subset G$ ($A$ is not necessarily a group) such that for all $a\in A$, there does not exist some non-identity $x\in G$ with both $xa, x^{-1} a \in A$ (for the same $x$). How can we evaluate the maximal cardinality of $A$?
$A$ will of course depend on the structure of the group itself, not just the cardinality of it, so I do not expect some answer in terms of the size of the group but rather some method of obtaining $|A|$ through operations directly on $G$.