Subgroups of p-groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:16:39Z http://mathoverflow.net/feeds/question/57104 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57104/subgroups-of-p-groups Subgroups of p-groups Martin David 2011-03-02T11:23:23Z 2011-04-18T17:08:37Z <p>If \$G\$ is a (non-abelian) \$p\$-group, \$|G|=p^n\$, \$n>3\$, then it is elementary that \$G\$ contains a (normal) abelian subgroup of \$p^2\$. It is also true that \$G\$ necessarily contains a <strong>normal abelian subgroup</strong> of order \$p^3\$ (<em>Group Theory - W. R. Scott</em>).</p> <p><strong>1)</strong> What is the largest possible value of \$m\$ such that <strong>any</strong> non-abelian group of order \$p^n\$ contains a <strong>normal abelian subgroup</strong> of order \$p^m\$? </p> <p><strong>2)</strong> What is the latgest possible value of \$m\$ such that <strong>any</strong> non-abelian group of order \$p^n\$ contains an <strong>abelian subgroup</strong> of order \$p^m\$? </p> <p>[Please suggest references.]</p> http://mathoverflow.net/questions/57104/subgroups-of-p-groups/57138#57138 Answer by mt for Subgroups of p-groups mt 2011-03-02T17:22:18Z 2011-03-08T15:08:07Z <p>I've been asked to post the following as an answer, although it does not answer either of your questions (i.e. it does not provide the largest \$m\$).</p> <p>Here are some suggested references: G A Miller, On the number of abelian subgroups.. in Messenger Math 36 (1906/7). <a href="http://www.jstor.org/stable/1996255" rel="nofollow">SC Dancs, Abelian subgroups of finite \$p\$-groups in Trans AMS 169 (1972)</a>. Miller shows a group of order \$p^n\$ has a normal abelian subgroup of order \$p^m\$ for some \$m\$ such that \$m (m+1)/2 \geq n\$. The inequality is correct. Huppert's Endliche Gruppen book is cited there as an alternative proof of Miller's paper (what I remember of that paper is that it is very hard to read).</p> <p>Edit: a better reference is Zassenhaus's book `The Theory of Groups', IV.3.4. There you find a simple argument for the lower bound above. It's clear from his proof that the lower bound can be improved if you have control of the number of generators of a maximal normal abelian subgroup, for example in the case that the big group is regular.</p> http://mathoverflow.net/questions/57104/subgroups-of-p-groups/61960#61960 Answer by Geoff Robinson for Subgroups of p-groups Geoff Robinson 2011-04-16T21:19:04Z 2011-04-18T17:08:37Z <p>George Glauberman and also Jon Alperin and George Glauberman together have written papers on this topic in recent years. One example is: "A note on abelian subgroups of p-groups." Groups St. Andrews 2005. Vol. 2, 445–447, London Math. Soc. Lecture Note Ser., 340, Cambridge Univ. Press, Cambridge, 2007.</p>