A new solvability criterion for finite groups

Abstract

Abstract. In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if for all conjugacy classes C and D of G, there exist x ∈ C and y ∈ D for which x, y is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two integers a and b which represent orders of elements in G and for all elements x, y ∈ G with |x| = a and |y| = b, the subgroup x, y is nonsolvable.

Divisibility of binomial coefficients and generation of alternating groups

“…Dolfi, Guralnick, Herzog and Praeger remark in [7] that Condition (5) appears likely to hold with asymptotic density 1. We show the following.…”

“…See for example [3,5,8,16]. Dolfi, Guralnick, Herzog and Praeger first ask Question 1.4 in [7,Section 6]. These authors conjecture that the analogue of Condition (5) holds for all but finitely many simple groups of Lie type, but point out that the corresponding statement for alternating groups occasionally fails.…”

Conjugacy classes, characters and products of elements

“…Actually, a weaker sufficient condition for nilpotence is known: G is nilpotent if and only if for every pair of primes p and q and every pair of elements with x a p ‐element and y a q ‐element, x commutes with some conjugate of y (see Corollary E of ), but the proof of this result depends on the classification of finite simple groups.…”

“…By induction ∕⟨ ⟩ has a Sylow -subgroup as a direct factor whence so does . □ Actually, a weaker sufficient condition for nilpotence is known: is nilpotent if and only if for every pair of primes and and every pair of elements , ∈ with a -element and a -element, commutes with some conjugate of (see Corollary E of [4]), but the proof of this result depends on the classification of finite simple groups. Now, we are ready to prove our strong form of Theorem A.…”

On Certain Weak Engel-Type Conditions in Groups

“…It is a well-known fact that nilpotency is bigenetic in the class of all finite groups, and in [13], Thompson also showed that a finite group is solvable when every pair of its elements generates a solvable group. More recently, Dolfi et al [3] proved that solvability and nilpotency of finite groups are ensured by weaker conditions: a finite group G is solvable (nilpotent) if for every pair of elements a b ∈ G there exists an element g ∈ G such that the subgroup a g b is solvable (nilpotent).…”