A solvable version of the Baer–Suzuki theorem

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.…”

Criteria for solvable radical membership via p-elements

“…(a2) Guralnick and Wilson [18] (2000): G is solvable if and only if the proportion of pairs (x, y) ∈ G × G (with x = y) for which x, y is solvable is at least 11/30. (a3) The first author [15] (2010) (and (independently) Gordeev, Grunewald, Kunyavskii, and Plotkin in [12] (2010)): G is solvable if and only if x, x y is solvable for all x, y ∈ G.…”

On the Fitting height of a finite group

“…Recently Guest [5] obtained a solvable version of the Baer-Suzuki theorem by proving that an element x a G of prime order p d 5 is contained in a normal solvable subgroup if and only if 3x; x g 4 is solvable for all g a G. The result depends only on Thompson's classification of minimal simple groups [6] and thus is independent of the full classification of finite simple groups. One corollary of Guest's result is that G is solvable if and only if every pair of conjugate elements generate a solvable subgroup (see also [2]).…”

“…Let G be a finite group in which each pair of conjugate elements generate a solvable subgroup whose Fitting height is at most h. It follows from [3] or [5] that G is solvable. Suppose that hðGÞ d h þ 1.…”