commutator subgroups and isomorphic - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:31:25Z http://mathoverflow.net/feeds/question/94193 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94193/commutator-subgroups-and-isomorphic commutator subgroups and isomorphic ali tavakoli 2012-04-16T08:38:43Z 2012-04-17T09:09:53Z <p>Let G and H be finitely generated groups. Let G' = [G,G] and H' = [H,H] be the corresponding derived groups (commutator subgroups) of G and H. we know that there exist groups G and H such that G' is isomorphic to H' and G/G' is isomorphic to H/H' but G and H are not isomorphic. with which additional condition on G,H can we obtain G and H are isomorphic? (for example residually finite, FC-group, m-generators with n-expanent,...)</p> http://mathoverflow.net/questions/94193/commutator-subgroups-and-isomorphic/94283#94283 Answer by Simon Lentner for commutator subgroups and isomorphic Simon Lentner 2012-04-17T09:09:53Z 2012-04-17T09:09:53Z <p>Wei Zhou's answer already gives a very good example and I agree with the above comments. <strong>However</strong> for a "way out" - I want to give you two classes of obtacles, if you additionally remove the first, you're on your way to a well-known equivalence, which is weaker than isomorphy but helps e.g. in $p$-groups.</p> <p><strong>1) In my opinion the serious thing:</strong> You loose knowledge of the commutator map $G/G'\times G/G'\rightarrow G'/G^{(2)}$ and only preserve THAT elements appear as commutators. </p> <p>Take as <strong>example</strong> the <strong>extraspecial groups</strong> $p^{2n+1}_\pm$ (such as the direct product with centers identified $D_4\ast\ldots\ast D_4$) compared to e.g. these one: $p_\pm^{2\cdot 1+1}\times \mathbb{Z}_p^{2(n-1)}$ (such as $D_4\times \mathbb{Z}_2^{2(n-1)}$). Both are central/stem-extensions of $\mathbb{Z}_p^{2n}$ by $\mathbb{Z}_p$, but the second is a lot "more commutative" ;-)</p> <p><strong>2) A lot more tricky to detect but maybe not-so-serious:</strong> Even if the commutator maps match, it remains unclear how elements powered-up/"fused" to the divided-out commutators.</p> <p>Best <strong>examples</strong> are certainly the different extraspecials $p_+^{2\cdot n+1}$ vs. $p_-^{2\cdot n+1}$, especially $D_4$ vs. $Q_8$ (as Zhou said), that can only be distinguished by how many elements powering to the central commutator. </p> <p>The latter behaviour is sometimes described as the groups being <strong>isoclinic</strong> (there are different definitions!) and this is also responsible e.g. for the three different nonabelian isomorphy types in order $2^n$. It is fairily mild and used e.g. in the classification efforts on $p$-groups. Most prominent examples you get from <strong>different Schur covers</strong> of a group being isoclinic (again $D_4/Q_8$ but also the different 2-covers of $S_n$). In this situation by construction all isoclinics are <strong>isomorphic as grouprings</strong> $k[D_4]\cong k[Q_8]$ and hence have the <strong>same representation theory</strong>, although I have no sources confirming this for nonabelian extensions!!</p> <p><strong>Hope that helps ;-)</strong> </p>