The involution width of finite simple groups

Abstract

In this paper we measure how efficiently a finite simple group G is generated by its elements of order p, where p is a fixed prime. This measure, known as the p-width of G, is the minimal k ∈ N such that any g ∈ G can be written as a product of at most k elements of order p. Using primarily character theoretic methods, we sharply bound the p-width of some low rank families of Lie type groups, as well as the simple alternating and sporadic groups.

Surjective word maps and Burnside’s $$p^aq^b$$ p a q b theorem

“…There is also considerable literature on the case of involutions, see e.g. [36] and the references therein. These imply results like Theorem 2 with longer products x N 1 x N 2 .…”

On the Orbital Diameter of Groups of Diagonal Type

“…A group is strongly real if and only if any of its elements can be expressed as a product of two involutions. Theorem 2.3 ([8,10,12,13,17,23,25,27,28]). Let G be a non-abelian finite simple group.…”

The p-width of the alternating groups

“…Firstly assume that g 1 g 2 g 3 ∈ A n and consider the sub-element g 1 g 2 ∈ S n . If g 1 g 2 satisfies either condition (8) or (9) then the conclusion of the lemma follows. We therefore assume that this is not the case and so we must have insufficient letters i.e.…”

“…In particular, [3,Thm. 10.2] gives a sufficient condition for three copies of a given conjugacy class to cover A n (see Remark 1.1 and Appendix A) In the more general case of non-abelian finite simple groups, the problem of the 2-width (more commonly known as the involution width) has been solved by this author [9]: the involution width of any finite simple group is at most four, and this bound is sharp [9,Thm. 1].…”

“…1]. Specifically, the involution width of A n is two if n ∈ {5, 6, 10, 14} and three otherwise [9,Thm. 2.3].…”