Structure of finite groups with four conjugacy class sizes of certain elements

Let G be a finite group. If the set of conjugacy class sizes of primary and biprimary elements is {1,m,n,mk}, where (m,n) = 1 and k > 1 is a proper divisor of n, we prove that G is solvable with an abelian Hall π(m)-subgroup. Moreover, G∕Z(G) is either a Frobenius group or a 2-Frobenius group of order mn. We also give a detailed structure description of G∕Z(G), which is a generalization of [9 Chen, R. F., Zhao, X. H. (2016). A criterion for a group to have nilpotent p-complements. Monatsh. Math. 179(2): 221–225.[Crossref], [Web of Science ®] , [Google Scholar], Theorem A].