Let G be a finite group. We extend Alan Camina’s theorem on conjugacy classes sizes which asserts that if the conjugacy classes sizes of G are {1, p a , q b , p a q b }, where p and q are two distinct primes and a and b are integers, then G is nilpotent. We show that let G be a group and assume that the conjugacy classes sizes of elements of primary and biprimary orders of G are exactly {1, p a , n,p a n} with (p, n) = 1, where p is a prime and a and n are positive integers. If there is a p-element in G whose index is precisely p a , then G is nilpotent and n = q b for some prime q ≠ p.
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机译：令G为有限群。我们扩展了共轭类大小的Alan Camina定理，该定理断言，如果G的共轭类大小为{1，p a ，q b ，p a q b }，其中p和q是两个不同的质数，而a和b是整数，则G是幂等的。我们表明，让G为一个组，并假设G的主要和双主要阶元素的共轭类大小恰好是{1，p a ，n，p a （}，其中p为质数，a和n为正整数。如果G中存在一个p元素，其索引恰好是p a ，则对于某些素数q≠p，G是幂等的，并且n = q b 。
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