Does probability of a derangement go up under passing to subgroups? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-25T09:50:15Z http://mathoverflow.net/feeds/question/103009 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103009/does-probability-of-a-derangement-go-up-under-passing-to-subgroups Does probability of a derangement go up under passing to subgroups? David Speyer 2012-07-24T15:12:29Z 2012-07-24T15:54:53Z <p>This is prompted by my attempts to work on <a href="http://mathoverflow.net/questions/102751/a-mixing-property-for-finite-fields-of-characteristic-2/" rel="nofollow">this question</a>. Let \$H \subset G \subseteq S_d\$ be transitive permutation groups. Recall that an element of \$S_d\$ is called a derangement if it has no fixed points.</p> <blockquote> <p>Is the proportion of derangements in \$H\$ always greater than in \$G\$?</p> </blockquote> <p>If \$H\$ doesn't have to be transitive, then the answer is "no"; just let \$H\$ be trivial. But a quick sampling of examples with \$G\$ and \$H\$ both transitive doesn't turn up any counterexamples.</p> <p><b>UPDATE</b> Never mind. \$A_4\$ inside \$S_4\$, the probability of a derangement goes down from \$3/8\$ to \$1/4\$.</p>