Given two (finite) groups $G$ and $D$ and their commutators $G'$ and $D'$, is it true that if $G/(G') \cong D/(D') $ and $G' \cong D'$, then $G \cong D$?
2026-03-25 07:48:52.1774424932
Isomorphism of quotients over isomorphic commutators
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No. Take $G=D_8$ and $D=Q_8$. Here $G'\cong D' \cong \Bbb Z_2$ and $G/G' \cong D/D' \cong V_4$, where $V_4$ is the Klein four-group.