The Baer–Suzuki width of a complete class of finite group is finite

Let X \mathscr {X} be a nonempty class of finite groups closed under taking subgroups, homomorphic images, and extensions. According to Gordeev, Grunewald, Kunyavskiĭ, and Plotkin, the Baer–Suzuki width B S ( X ) \mathrm {BS}(\mathscr {X}) of X \mathscr {X} does not exceed a nonnegative integer m m if, in any finite group G G , the largest normal X \mathscr {X} -subgroup coincides with the set of elements x x such that every m m elements conjugate to x x generate an X \mathscr {X} -subgroup. If there are no m m for which B S ( X ) ≤ m {\mathrm {BS}(\mathscr {X})\leq m} , then by definition B S ( X ) = ∞ {\mathrm {BS}(\mathscr {X})=\infty } . In the paper, it is proved that