  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.
