Let $S$ and $T$ be non-empty subsets of a group $G$. As usual $ST=\{st : s \in S, t\in T\}$. What can be said of the subgroups $\langle ST\rangle$ and $\langle TS\rangle$? For example if the identity $1 \in S \cap T$, then it is easy to see that $\langle ST\rangle$ = $\langle TS\rangle$. Also, if S and T are both singletons, then $\langle ST\rangle$ and $\langle TS\rangle$ are conjugated, since $ST$ and $TS$ are conjugated sets in this case. Are $\langle ST\rangle$ and $\langle TS\rangle$ always conjugated?
Products of sets in a group
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group-theory
finite-groups