[Paper Review] On the maximum orders of elements of finite almost simple groups and primitive permutation groups

Simon D. Guest, Joy Morris+2 more|ePrints Soton (University of Southampton)|Jan 22, 2013

Finite Group Theory Research20 references4 citations

TL;DR

This paper establishes sharp upper bounds on the maximum element order (meo) in finite almost simple groups and primitive permutation groups, showing that for non-alternating groups, meo(G) ≤ m(T) with finitely many exceptions, and meo(G) ≤ m(T)/4 except for 12 exceptional simple groups. It further classifies all primitive permutation groups of degree n that contain elements of order at least n/4, identifying their socle structures and providing explicit bounds via group-theoretic analysis of element orders and permutation representations.

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.

Motivation & Objective

• To determine upper bounds for the maximum order of an element in finite almost simple groups in terms of the minimal degree m(T) of a faithful permutation representation of the socle T.
• To investigate when the maximum element order meo(G) is bounded by a constant fraction of m(T), particularly m(T)/4, and identify exceptional cases.
• To classify all primitive permutation groups of degree n that contain elements of order at least n/4, identifying their socle structure and permutation action.
• To extend results from symmetric and alternating groups to other finite simple groups and their automorphism groups, especially classical groups.
• To provide explicit bounds and exceptions for meo(Aut(T)) in terms of m(T), with special attention to PSL_d(q) and PSU_4(q) families.

Proposed method

• The authors use known bounds on meo(Aut(T)) for finite simple groups T, particularly classical groups, derived from cycle structure and field size.
• They compare meo(Aut(T)) with m(T), the minimal degree of a faithful permutation representation of T, using asymptotic estimates and explicit calculations.
• For PSU_d(q) groups, they derive asymptotic inequalities showing that meo(Aut(T)) < m(T)^{3/4} for large q, except for finitely many cases.
• They apply group-theoretic techniques to analyze the structure of primitive permutation groups of product action type, especially wreath products and socle actions.
• They use computational verification and case analysis to determine the maximal ℓ_T such that meo(G) ≥ n/4 for G of PA type with socle T^ℓ.
• They classify the possible socle types of primitive groups G with meo(G) ≥ n/4, identifying natural actions of Alt(m)^ℓ and PSL_d(q)^ℓ on tuples of subspaces or points.

Experimental results

Research questions

• RQ1 What is the best possible upper bound for meo(G) in terms of m(T) for finite almost simple groups with non-alternating socle T?
• RQ2 For which finite non-abelian simple groups T is meo(Aut(T)) > m(T)/4, and how many such exceptions exist?
• RQ3 Which primitive permutation groups of degree n contain elements of order at least n/4, and what are their socle structures?
• RQ4 Can the bound meo(G) ≤ m(T) be improved to meo(G) ≤ m(T)/4 for all but finitely many non-alternating, non-PSL_d(q) groups?
• RQ5 What is the maximal ℓ such that the wreath product T ≀ Sym(ℓ) admits an element of order at least n/4, where n = m^ℓ?

Key findings

• For all but finitely many non-alternating finite simple groups T, the maximum element order in Aut(T) is at most m(T), with equality holding for all but two (d,q) pairs in PSL_d(q).
• For all but 12 exceptional simple groups T, the bound improves to meo(Aut(T)) < m(T)/4, with the exceptions listed in Table 1.
• For T = PSU_4(q), meo(Aut(T)) = q^3 + 1 grows faster than m(T)^{3/4 - ε} for any ε > 0 and large q, showing the bound is sharp up to ε.
• All primitive permutation groups G of degree n with meo(G) ≥ n/4 have socle isomorphic to Alt(m)^ℓ, PSL_d(q)^ℓ, or an elementary abelian group, with specific actions on tuples of subspaces or points.
• The maximal ℓ_T such that G = T ≀ Sym(ℓ) has meo(G) ≥ n/4 is finite for each T, with ℓ_T = 4 for Alt(7), ℓ_T = 3 for Alt(5), and ℓ_T = 1 for HS and other sporadic groups.
• Explicit values of ℓ_T and corresponding m such that n = m^ℓ and meo(G) ≥ n/4 are tabulated in Table 6 for 22 simple groups, including Alt(m), M11, M12, M22, M23, M24, and various PSL_d(q), PSU_d(q), and PSp groups.