Criteria for solvable radical membership via p-elements

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.