The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of finite CP-groups yields a detailed description of profinite groups with that property. Then we deal with two generalizations of profinite CP-groups. Theorem 1.2. A profinite group G is virtually pro-p for some prime p if and only if for each nontrivial x in G there is a prime p (depending on x) such that the centralizer of x is virtually pro-p. Theorem 1.3. Let G be a profinite group in which each element has either finite or prime power (possibly infinite) order. Then G is either torsion or virtually pro-p for some prime p.

中文翻译：

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许多元素具有质数幂阶的有限群

有限群和局部有限群的结构是众所周知的，其中每个元素都具有质数幂阶（CP-群）。在本文中，我们注意到，将我们早期的结果与有限 CP 群结构的可用信息相结合，可以对具有该性质的超限群进行详细描述。然后我们处理profinite CP-groups的两个推广。定理 1.2。当且仅当对于 G 中的每个非平凡 x 存在一个素数 p（取决于 x），使得 x 的中心化项实际上是 pro-p 的，一个超限群 G 对某个素数 p 实际上是 pro-p 的。定理 1.3。设 G 是一个有穷群，其中每个元素都具有有限阶或素数阶（可能是无限阶）。那么 G 要么是扭转，要么是对某个素数 p 的近似 pro-p。