Characterizations of the solvable radical

Abstract

Abstract. We prove that there exists a constant k with the property: if C is a conjugacy class of a finite group G such that every k elements of C generate a solvable subgroup, then C generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take k = 4. We also present proofs that do not use the Classification Theorem. The most direct proof gives a value of k = 10. By lengthening one of our arguments slightly, we obtain a value of k = 7.

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

“…Remark 1.8. R. Guralnick informed us that Theorems 1.4 and 1.3 were independently proved in forthcoming works by Guest, Guralnick, and Flavell [16,17,8]. Flavell [8] reduced k to 7 with a proof which does not rely on CFSG.…”

“…R. Guralnick nous a informé que les Théorèmes 0.3 et 0.2 ont été démontrés indépendamment dans des travaux à paraître par Guest, Guralnick et Flavell [16,17,8]. De plus, Flavell [8] a réduit le nombre k à 7, la démonstration n'utilisant pas la classification.…”

A solvable version of the Baer–Suzuki theorem

“…So Theorem A implies that there exists g ∈ G such that x, x g is not solvable, and thus G is not a minimal counterexample. The version of the theorem for linear groups follows from the finite group version using a standard argument (see [FGG,Corollary 1.2] for example).…”

Multilinear commutators in residually finite groups

“…The proof also relies on the classification of finite simple groups as well as on the Lie-theoretic techniques that Zelmanov created in his solution of the Restricted Burnside Problem. We also use recent result, essentially due to Flavell, Guest and Guralnick, that an element a of a finite group G belongs to F h (G) if and only if every 4 conjugates of a generate a soluble subgroup of Fitting height at most h [6]. It is the necessity to use this result that accounts for the difference between the constants 896 in Theorem 1.5 and 68 in the case of simple commutators [x, y] [23].…”