20.81 (2022)

Does there exist a class $\mathfrak{X}$ of finite groups satisfying the following conditions:
$\qquad$ (1) $\mathfrak{X}$ is closed with respect to taking subgroups and homomorphic images;
$\qquad$ (2) the product of two normal $\mathfrak{X}$-subgroups of an arbitrary group is always an $\mathfrak{X}$-group;
$\qquad$ (3) for every positive integer $n$, there exist a finite group and its conjugacy class $D$ such that any $n$ elements of $D$ generate an $\mathfrak{X}$-subgroup, whereas $\langle D \rangle \notin \mathfrak{X}$.

It is known that there are no classes $\mathfrak{X}$ satisfying (1)–(3) that are closed with respect to extensions (D. O. Revin, Algebra i Analiz, 37, no. 1 (2025), 141–176 (Russian)).