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Variations on a theorem by Alan Camina on conjugacy class sizes

2005; Elsevier BV; Volume: 296; Issue: 1 Linguagem: Inglês

10.1016/j.jalgebra.2005.06.031

1090-266X

Antonio Beltrán, María José Felipe,

Limits and Structures in Graph Theory

Let G be a finite group. We extend Alan Camina's theorem on conjugacy class sizes which asserts that if the conjugacy class sizes of G are exactly {1,pa,qb,paqb} for two primes p and q, then G is nilpotent. If we assume that G is solvable, we show that when the set of conjugacy class sizes of G is {1,m,n,mn} with m and n arbitrary positive integers such that (m,n)=1, then G is nilpotent and m=pa and n=qb for two primes p and q.