abelian subgroups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:11:50Z http://mathoverflow.net/feeds/question/117402 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117402/abelian-subgroups abelian subgroups darya 2012-12-28T12:14:24Z 2012-12-28T23:08:38Z <p>Have the groups "PSL(n,q)" and "PSL(n,q).f ", the same maxiaml abelian subgroups or not?(where "PSL(n,q).f " is the extension of PSL(n,q) by the field automorphism of it) Is there any counterexample for my question for example for some n,q?</p> http://mathoverflow.net/questions/117402/abelian-subgroups/117456#117456 Answer by Peter Mueller for abelian subgroups Peter Mueller 2012-12-28T23:08:38Z 2012-12-28T23:08:38Z <p>\$n=2, q=4\$ is a counterexample: \$PSL(2,4)=A_5\$, the alternating group of degree \$5\$, which doesn't have abelian subgroups of order \$\ge6\$, and \$PSL(2,4)\rtimes Aut(GF(4))=S_5\$, which has an abelian subgroup of order \$6\$. These are probably the only counterexamples.</p> <p>Suggestion: Try to consider preimages of abelian groups in \$SL(n,q)\$, while they need not be abelian, maybe one can still play with Schur's Lemma and such. Also note that \$SL(n,p^e)\rtimes Aut(GF(p^e))\$ embeds into \$GL_{ne}(p)\$, which might be helpful.</p>