Solvabilizer Numbers of Finite Groups

Abstract

Consider a nonsolvable finite group G, where R(G) represents the solvable radical of G. For any element x in G, the solvabilizer of x in G, denoted by Sol_G(x), is defined as the set of all elements y in G such that the subgroup generated by x and y is solvable. Notably, the entirety of G can be expressed as the union over all x in G\R(G) of their respective solvabilizers: $G = \cup_{x\in G\R(G)} Sol_G(x). A solvabilizer covering of G is characterized by a subset X of G\R(G) such that G= \cup_{x\in X} Sol_G(x). The solvabilizer number of G is then defined as the minimum cardinality among all solvabilizer coverings of G. This paper delves into the exploration of the solvabilizer number for diverse nonsolvable finite groups G, shedding light on the interplay between solvability and the structure of these groups.