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

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. 2000 Mathematics Subject Classification. 20B15, 20H30.

Alternating and Symmetric Groups With Eulerian Generating Graph

“…Since g has no two cycles of same length, the proof of [10,Corollary 2.7] yields that |g| divides n = (q Since |g| is odd, we must have r = 2. Moreover, as |g| is a power of an odd prime, Zsigmondy's theorem yields that d = 2.…”

“…A maximal subgroup H of G containing g is clearly transitive, and is either imprimitive or primitive. If H is primitive then by [10 − 1)). We claim that d > 2.…”

A CHARACTERIZATION OF SOME PGL(2, q) BY MAXIMUM ELEMENT ORDERS

“…Let π e (G) = {|g| : g ∈ G} and meo(G) = max{|g| : g ∈}; that is, π e (G) is the set of orders of elements of G and meo(G) is the maximum element order of G (see [11]). A group G is called a 2-Frobenius group if G has a normal series 1 ⊆ H ⊆ K ⊆ G such that K and G/H are Frobenius groups with kernels H and K/H respectively.…”

“…In addition, the two largest element orders of simple groups of Lie type of odd characteristic are listed in [17]. In [11], Guest, Morris, Praeger and Spiga determine the upper bounds of meo(S) for S a finite almost simple group. Moreover, these results are applied to determine the primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n/4.…”

“…Our study is motivated by [11] and [17], which contain the maximum element orders of simple groups of Lie type. It is well known that meo(P GL(2, q)) = q + 1, where q = p n is a prime power, and meo(P SL(2, p)) = p with p > 2 a prime (see [11] and [17] respectively or see Dickson's book [9]). Our first result shows that the maximum element orders of P GL(2, p) together with their orders in turn determine P GL (2, p).…”