Groups with no perfect subgroups -- terminology? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:50:57Z http://mathoverflow.net/feeds/question/41862 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41862/groups-with-no-perfect-subgroups-terminology Groups with no perfect subgroups -- terminology? Jeff Strom 2010-10-12T01:57:13Z 2010-10-12T11:25:34Z <p>Finite groups are solvable if they have no nontrivial perfect subgroup. But I am sure that for infinite groups, the two notions diverge. Is there standard terminology for groups with no perfect subgroups?</p> http://mathoverflow.net/questions/41862/groups-with-no-perfect-subgroups-terminology/41863#41863 Answer by Mark Sapir for Groups with no perfect subgroups -- terminology? Mark Sapir 2010-10-12T02:52:52Z 2010-10-12T11:25:34Z <p>In the infinite case, there is a close notion of "locally indicable group", i.e. a group where every finitely generated subgroup maps onto \$\mathbb Z\$ (see, for example, <a href="http://www.jstor.org/pss/2699673" rel="nofollow">this paper</a>). Locally indicable groups are left (right) orderable, hence important. Note that in that notion, not all subgroups are considered but only finitely generated, and "non-perfect" is replaced by a stronger property "maps onto \$\mathbb Z\$". But in the finite case all subgroups are finitely generated, and "maps onto \$\mathbb Z\$" is an infinite analog of "maps onto a finite cyclic group" (= non-perfect). So "locally indicable" is possibly the infinite analog of the property you consider. </p> <p>Update: The groups without perfect subgroups are called hypoabelian. See <a href="http://planetmath.org/encyclopedia/SDGroup.html" rel="nofollow"> this text. </a></p>