Let $G$ be a finite group. If $a = bab$, is it true that $b^{2} = e$ Let $G$ be a finite group and let $a,b \in G$. If $a = bab$, is it true that $b^{2} = e$. If not, find a counterexample. It is clear that if $a = b...--prophetes.ai

Let $G$ be a finite group. If $a = bab$, is it true that $b^{2} = e$ Let $G$ be a finite group and let $a,b \in G$. If $a = bab$, is it true that $b^{2} = e$. If not, find a counterexample. It is clear that if $a = bab$ and $b^{2} = e$ are both true, then $ab = ba$. However, there exist groups (namely non-Abelian ones) with elements such that $ab \neq ba$. However, I am having trouble finding a non-Abelian group with elements such that $a = bab$, but $ab \neq ba$. How does one solve this problem?

No. For example, in $Q_8$ we have $jij=jk=i$.