Arithmetical Conditions on the Conjugacy Class Numbers of a Finite Group

Solvability of normal subgroups and $G$-class sizes

“…In particular, the index in N of any r -element of N (notice that N ⊆ C G (w)) is an r -number as claimed, and by an elementary result (see for instance [6]), N factorizes as a direct product of an r-group R and an r -group H. Notice that H G, so by applying induction, H is solvable, whence N is solvable too.…”

Finite groups with three conjugacy class sizes of primary and biprimary elements

“…For instance, a classical result due to Itô [7] is that a group G is nilpotent if its set of conjugacy class sizes is {1, m} for some fixed integer m. He also proved [8] that G is solvable if the set of its conjugacy class sizes is {1, m, n} for positive integers m and n. In [5], Camina proved the following: Theorem 1.1 ([5,Theorem 3] On the other hand, some authors replace conditions for all conjugacy classes by conditions referring only to some conjugacy classes to investigate the structure of a group. For instance, Baer [2] proved that a group is solvable if its primary elements have prime power conjugacy class sizes.…”

The structure of finite groups with three class sizes

“…For instance, it was first proved in [13] that if the conjugacy class sizes of G are f1; m; ng with m; n > 1 coprime, then G=ZðGÞ is a Frobenius group and the inverse image in G of the kernel and a complement are abelian. Also, Camina determined in [4] the structure of a group whose class sizes are f1; p a ; p a q b g for distinct primes p and q (in this case solvability is immediate).…”