Timeline for commutator subgroups and isomorphic

6 events

when toggle format	what		by	license	comment
Apr 17, 2012 at 9:09	answer	added	Simon Lentner		timeline score: 4
Apr 16, 2012 at 13:32	comment	added	Wei Zhou		I think $Q_8$ and $D_8$ are such examples.
Apr 16, 2012 at 13:14	comment	added	Wei Zhou		I think that there exist finite groups $G, H$ with $G'\cong H'$ and $G/G' \cong H/H'$, but $G, H$ are not isomorphic. So all your additional conditions, as finiteness conditions, are not enough.
Apr 16, 2012 at 10:10	comment	added	HJRW		Sorry, there's a typo - the commutator subgroup is isomorphic to $\mathbb{Z}^2$.
Apr 16, 2012 at 9:00	comment	added	HJRW		Certainly residual finiteness is not enough. For instance, consider a semi-direct product $\Gamma_\phi=\mathbb{Z}^2\rtimes_\phi\mathbb{Z}$, where the eigenvalues of $\phi$ are real and distinct. Then $\Gamma_\phi$ is residually finite with $[\Gamma_\phi,\Gamma_\phi]\cong\mathbb{Z}$ and the abelianisation isomorphic to $\mathbb{Z}$ but $\Gamma_\phi$ and $\Gamma_\psi$ are isomorphic if and only if $\phi$ and $\psi$ are $\mathbb{Z}$-conjugate.
Apr 16, 2012 at 8:38	history	asked	ali tavakoli	CC BY-SA 3.0