On the maximum orders of elements of finite almost simple groups and primitive permutation groups

Guest, Simon; Morris, Joy; Praeger, Cheryl; Spiga, Pablo

doi:10.1090/S0002-9947-2015-06293-X

On the maximum orders of elements of finite almost simple groups and primitive permutation groups
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by Simon Guest, Joy Morris, Cheryl E. Praeger and Pablo Spiga PDF

Trans. Amer. Math. Soc. 367 (2015), 7665-7694

Abstract:

We determine upper bounds for the maximum order of an element of a finite almost simple group with socle $T$ in terms of the minimum index $m(T)$ of a maximal subgroup of $T$: for $T$ not an alternating group we prove that, with finitely many exceptions, the maximum element order is at most $m(T)$. Moreover, apart from an explicit list of groups, the bound can be reduced to $m(T)/4$. These results are applied to determine all primitive permutation groups on a set of size $n$ that contain permutations of order greater than or equal to $n/4$.

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