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

Let X 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 ) BS(X) of X X does not exceed a nonnegative integer m m if, in any finite group G G , the largest normal X X -subgroup coincides with the set of elements x x such that every m m elements conjugate to x x generate an X X -subgroup. If there are no m m for which B S ( X ) ≤ m {BS(X)≤ m} , then by definition B S ( X ) = ∞ {BS(X)=∞ } . In the paper, it is proved that