If $G$ and $H$ be groups with $\mathbb{Z}G \simeq \mathbb{Z}H$ then $\frac{G}{G^{'}}\simeq \frac{H}{H^{'}}$.
It will be great if you help me with this. Any hint or guidance will be great. Thanks.
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2See http://math.stackexchange.com/questions/275249/quotient-of-ideal-in-group-ring-is-isomorphic-to-abelianization. – Dietrich Burde Jan 07 '15 at 10:57
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@Eric, $G'$ is standard notation for the derived subgroup, aka $[G,G]$. The quotient $G/G'$ is called the abelianization, sometimes denoted $G^{\rm ab}$ (cuz every abelian image of $G$ is a quotient of $G^{\rm ab}$). – anon Jan 07 '15 at 11:09
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I figured it was something standard but I can't just go around searching "G'" online :D – Eric Stucky Jan 07 '15 at 11:10
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@EricStucky, my fault. – Jihad Jan 07 '15 at 11:32
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@DietrichBurde: Is it clear a priori that an isomorphism of the group algebras would have to be compatible with the augmentation ideals (which I think would be needed to apply the approach your comment suggests)? – tracing Jan 14 '15 at 18:38