TITLE: Please help me, Group Theory. Prove $b^{33}=e$.
QUESTION [11 upvotes]: Let $G$ be a group and $a,b \in G$.
Prove that if $a^{2}=e$ and $ab^{4}a=b^{7}$, then $b^{33}=e$,
where $e$ is the identity of a group $G$.

REPLY [8 votes]: $$b^{49} = (b^7)^7 = (ab^4a)^7 = a (b^7)^4 a = a (a b^4 a)^4 a = b^{16}.$$