Question as in title: I have attempted this question by saying that the homomorpism that maps G to $Z_2$ is onto, $Z_2$ is abelian so G must be abelian.
2026-03-26 15:16:57.1774538217
$|G|= 2^n, o(g)=2~\forall g \in G, g \neq e$ show that G is abelian
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We have, for all $a,b\in G$, $$abba=e=baba$$ Hence $ab=ba$