SOLVABILITY OF FINITE GROUPS VIA CONDITIONS ON PRODUCTS OF 2-ELEMENTS AND ODD p-ELEMENTS

Abstract

We observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc. 12 (1911) has a nontrivial 2-element and an odd p-element, such that the order of their product is not divisible by either 2 or p. We also prove a solvability criterion involving conjugates of odd p-elements. Finally, we define, via a condition on products of p-elements with p -elements, a formation P p, p , for each prime p. We show that P 2,2 (which contains the odd-order groups) is properly contained in the solvable formation.2000 Mathematics subject classification: primaray 20D10; secondary 20E32.

Groups with some arithmetic conditions on real class sizes

“…(iii) Kaplan and Levy [35] prove that a finite group G is solvable if and only if for all odd primes p, for all p-elements x and 2-elements y in G, the subgroup x, x y is solvable.…”

A new solvability criterion for finite groups

“…A stronger result of this type was obtained recently by Kaplan and Levy in [KL,Theorem 4]. Their criterion involves only a limited 2generation within the conjugacy classes of elements of odd prime-power order.…”

Deciding FO-definability of regular languages

“…For a definition of solvable and unsolvable groups the reader is referred to [21]. Here, we only need the fact that any homomorphic image of a solvable group is solvable and the Kaplan-Levy criterion [16] (generalising Thompson's [26,Cor.3]) according to which a finite group G is unsolvable iff it contains three elements a, b, c, such that o G (a) = 2, o G (b) is an odd prime, o G (c) > 1 and coprime to both 2 and o G (b), and abc is the identity element of G.…”

“…| y ∈ {u, v} * . Then(16) implies that S ′ ⊆ S andδ B p MOD h(x) (s ′ ) = δ A p MOD x (s ′ ) ∈ S ′ ,for all s ′ ∈ S and x ∈ {u, v} * ,…”