It is a while since I read the proof, but I seem to remember that he computes the number for the Sylow 2-subgroup of the symmetric group and assumes that this is the worst case.

Indeed I have a paper with Laci Kovacs, Mike Newman and Cheryl Praeger where we show that you can have the integer below (2/3)2^n but no more.

Maybe I can answer my own question. Isbell has a lemma which asserts that if G is a transitive permutation group of degree 2^k, then that the proportion of elements of G having a fixed point is bounded by something tending to zero with k. This seems to be false: for instance, in a dihedral group of order 2^(k+1), 1/4 of the elements (plus one) have a fixed point.

]]>I’m afraid I must disagree. Science does not require belief; everything must be tested to destruction. Its failings are that sometimes people don’t notice and question some assumption right away.

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