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This is prompted by my attempts to work on this question. 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.

Is the proportion of derangements in $H$ always greater than in $G$?

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.

UPDATE Never mind. $A_4$ inside $S_4$, the probability of a derangement goes down from $3/8$ to $1/4$.

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closed as no longer relevant by Andreas Blass, Qiaochu Yuan, j.c., Will Sawin, Noah Stein Jul 24 '12 at 17:39

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I'm voting to close as no longer relevant, because, if I understand the MO software correctly, if the question remains open and (as I would expect) no answers (in the sense of the software) are given, then the question will keep reappearing on the front page from time to time. –  Andreas Blass Jul 24 '12 at 16:14
@Andreas Blass: I do not think this is the case. I never saw a question without an answer reappear just so. Those that reappear are those with answer(s) that are still in the 'unanswered' category, i.e. no accept answer and none with positive (or perhaps +2) score. (I once spend some time to check ca 2000 items in unanswered category looking for each question modified by the MO-user, ie having been bumped; very few have no answer and for a couple of those that currently have none I checked, with kind help of Felipe Voloch, that they used to have one.) –  quid Jul 24 '12 at 19:18