Solvabilizer numbers of finite groups

Consider a nonsolvable finite group [Formula: see text], where [Formula: see text] represents the solvable radical of [Formula: see text]. For any [Formula: see text], the solvabilizer of [Formula: see text] in [Formula: see text] is defined as [Formula: see text]. Notably, the entirety of [Formula: see text] can be expressed as the union over all [Formula: see text] of their respective solvabilizers: [Formula: see text]. A solvabilizer covering of [Formula: see text] is characterized by a set [Formula: see text] such that [Formula: see text]. The solvabilizer number of [Formula: see text] is then defined as the minimum cardinality among all solvabilizer coverings of [Formula: see text]. This paper delves into the exploration of the solvabilizer number for diverse nonsolvable finite groups [Formula: see text], shedding light on the interplay between solvability and the structure of these groups.