Criteria for solvable radical membership via p-elements

الأصل · EN

Guralnick, Kunyavskii, Plotkin and Shalev have shown that the solvable radical of a finite group G can be characterized as the set of all x∈ G such that <x,y> is solvable for all y∈ G. We prove two generalizations of this result. Firstly, it is enough to check the solvability of <x,y> for every p-element y∈ G for every odd prime p. Secondly, if x has odd order, then it is enough to check the solvability of <x,y> for every 2-element y∈ G.