virtual chain conditions in groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:29:05Z http://mathoverflow.net/feeds/question/58448 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58448/virtual-chain-conditions-in-groups virtual chain conditions in groups Colin Reid 2011-03-14T17:25:29Z 2011-05-23T21:22:13Z <p>In group theory, it's often very useful to know whether a family of subgroups (eg normal subgroups, Zariski-closed subgroups, ...) satisfies an ascending chain condition or a descending chain condition (that is, all ascending/descending chains in this family are finite). What I'm interested in is weaker 'virtual' chain conditions: a virtual DCC would be that given a sequence \$G_1 > G_2 > \dots\$ of subgroups (of some special kind) such that \$G_{i+1}\$ has infinite index in \$G_i\$ for all \$i\$, then the sequence must terminate. One can define virtual ACCs similarly.</p> <p>Does anyone know of work that has been done on conditions of this kind, either showing that a family of subgroups satisfies the conditions or deriving consequences from them? References for analogous conditions in other algebraic contexts would also be interesting.</p> http://mathoverflow.net/questions/58448/virtual-chain-conditions-in-groups/58449#58449 Answer by Simon Wadsley for virtual chain conditions in groups Simon Wadsley 2011-03-14T17:29:33Z 2011-03-14T17:29:33Z <p>The virtual DCC doesn't seem so different from the notion of Krull dimension \$1\$ that I explained in answer to this <a href="http://mathoverflow.net/questions/2525/different-definitions-of-the-dimension-of-an-algebra/2588#2588" rel="nofollow">http://mathoverflow.net/questions/2525/different-definitions-of-the-dimension-of-an-algebra/2588#2588</a> question. </p>