Abstract Héthelyi and Külshammer showed that the number of conjugacy classes k ⁢ ( G ) {k(G)} of any solvable finite group G whose order is divisible by the square of a prime p is at least ( 49 ⁢ p + 1 ) / 60 {(49p+1)/60} . Here an asymptotic generalization of this result is established. It is proved that there exists a constant c > 0 {c>0} such that, for any finite group G whose order is divisible by the square of a prime p, we have k ⁢ ( G ) ≥ c ⁢ p {k(G)\geq cp} .

中文翻译：

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根据素数限制有限群的类数

摘要 Héthelyi 和 Külshammer 表明，阶数可被素数 p 的平方整除的任何可解有限群 G 的共轭类数 k ⁢ ( G ) {k(G)} 至少为 ( 49 ⁢ p + 1 ) / 60 {(49p+1)/60}。这里建立了这个结果的渐近推广。证明存在常数 c > 0 {c>0} 使得对于阶数可被素数 p 的平方整除的任何有限群 G，我们有 k ⁢ ( G ) ≥ c ⁢ p {k( G)\geq cp}。