A commutator description of the solvable radical of a finite group

Abstract

Abstract. We are looking for the smallest integer k > 1 providing the following characterization of the solvable radical R.G/ of any finite group G: R.G/ coincides with the collection of all g 2 G such that for any k elements a 1 ; a 2 ; : : : ; a k 2 G the subgroup generated by the elements g; a i ga 1 i , i D 1; : : : ; k, is solvable. We consider a similar problem of finding the smallest integer`> 1 with the property that R.G/ coincides with the collection of all g 2 G such that for any`elements b 1 ; b 2 ; : : : ; b`2 G the subgroup generated by the commutators OEg; b i , i D 1; : : : ;`, is solvable. Conjecturally, k D`D 3. We prove that both k and`are at most 7. In particular, this means that a finite group G is solvable if and only if every 8 conjugate elements of G generate a solvable subgroup.

Baer–Suzuki theorem for the solvable radical of a finite group

“…Whenever possible, we maintain the notation of [10] which mainly follows [25,3]. In particular, we adopt the notation of [3] for twisted forms of Chevalley groups (so unitary groups are denoted by PSU n (q 2 ) and not by PSU n (q)).…”

“…Within past few years a lot of efforts have been made in order to describe the solvable radical of a finite group and to establish a sharp analogue of the Baer-Suzuki theorem with respect to the solvability property (see [6,7,9,10]). In particular, the following problem is parallel to the Baer-Suzuki result: Problem 1.2.…”

“…The reduction is fairly standard and follows the scheme of [9,10], so the rest of the paper is devoted to the outline of the proof of Theorem 2.1. We refer to the property stated in Theorem 2.1 as Property ( * ).…”

A solvable version of the Baer–Suzuki theorem

“…They have been obtained independently in [GGKP08b] and [GGKP08a], also using the Classification theorem.…”

From Thompson to Baer–Suzuki: A sharp characterization of the solvable radical

“…Whenever possible, we maintain the notation of [18] which mainly follows [39,11]. Let G(Φ, K ) be a Chevalley group where Φ is a reduced irreducible root system and K is a field.…”