Let G be a finite group and let $K=x^G$ be the conjugacy class of an element x of G. In this paper, it is proved that if N is a normal subgroup of G such that the coset $xN$ is the union of K and $K^{-1}$ (the conjugacy class of the inverse of x), then N and the subgroup $⟨K⟩$ are solvable. As an application, we prove that if there exists a natural number $n\ge 2$ such that $K^n=K\cup K^{-1}$, then $⟨K⟩$ is solvable.

中文翻译：

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正常子群的陪集和共轭类的幂

令G是一个有限群，令$钾=X^G$是共轭类的元素的X的G ^。本文证明，若N是G的正规子群，则陪集$XN$是K和${钾}^{-1}$（x的逆的共轭类），然后N和子群$⟨钾⟩$是可以解决的。作为一个应用，我们证明如果存在一个自然数$n\ge 2$ 以至于 ${钾}^n=钾\cup {钾}^{-1}$， 然后 $⟨钾⟩$ 是可以解决的。