On conjugacy class sizes of primary and biprimary elements of a finite group

Let m, n > 1 be two coprime integers. In this paper, we prove that a finite solvable group is nilpotent if the set of the conjugacy class sizes of its primary and biprimary elements is {1,m, n,mn}.

设\(m\)，\(n\gt1\)是两个互质的整数。在本文中，我们证明如果一个有限可解群的一次元和二元生成元的共轭类长的集合是\(\{1,m,n,mn\}\)，那么这个有限可解群是幂零的。