… for any x ∈ G the subgroup generated by x and y is solvable. … in the solvable radial of G if
and only if 〈P,x〉 is solvable for all … the Thompson-like characterization of the solvable radical. …

… prime order > 3 belongs to the solvable radical R(G) of a finite (… by g, xgx−1 is solvable. This
theorem implies that a finite (or … Shalev, Thompson-like characterization of radicals in groups …

… [26]) Let G be a finite group, let \(G_{{\mathcal {S}}}\) denote the soluble radical of G (ie the
largest soluble normal subgroup of G) and let \(x\in G\). Then, \(x\in G_{{\mathcal {S}}}\) if and …

… We prove that the solvable radical of a finite group G coincides with the set of elements y
having the following property: for any x ∈ G the subgroup of G generated by x and y is solvable. …

… soluble radical R(G) of a group G is the set of elements g such that the smallest normal subgroup
〈gG 〉 containing g is soluble… , Thompson-like characterizations of the solvable radical, …

… We give a survey of new characterizations of finite solvable groups and the solvable radi…
Characterization of the solvable radical Suppose we want to characterize the solvable radical …

… We call a criterion given in Theorem 1.1 the Thompson-like characterization of the solvable
radical. We also strengthen Theorem 1.1, showing that, if G is a finite group, and y1,y2 ∈ G …

… a finite group G belongs to the solvable radical RG/ of G if and … ] a Thompson-like
characterization of the solvable radical of … obtain a new characterization of the solvable radical RG/ …

… to prove a new characterization of the solvable radical R(G) … The solvable radical of any
finite group G consists of the … ] a Thompson-like characterization of the solvable radical of a …

… by the solvability property, … Thompson-like characterization of the solvable radical of finite
groups and finite dimensional Lie algebras [21] and Engel-like descriptions of finite solvable …