On profinite groups with the Magnus Property

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-06-04 DOI:10.1112/blms.70107

Claude Marion, Pavel Zalesskii

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Abstract

A group is said to have the Magnus Property (MP) if whenever two elements have the same normal closure then they are conjugate or inverse-conjugate. We show that a profinite MP group $G$ is prosolvable and any quotient of it is again MP. As corollaries, we obtain that a prime divisor of $| G |$ is 2, 3, 5 or 7, and the second derived subgroup of $G$ is pronilpotent. We also show that the inverse limit of an inverse system of profinite MP groups is again MP. Finally when $G$ is finitely generated, we establish that $G$ must in fact be finite.

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在无限群上用Magnus属性

如果两个元素具有相同的法向闭包，则它们是共轭的或反共轭的，则一个群具有马格努斯性质（MP）。证明了无限MP群G$ G$是可解的，并且它的任何商都是MP。作为推论，我们得到了|G|$ |G|$的一个素数是2、3、5或7，并且得到了G$ G$的第二个导出子群是幂幂的。我们还证明了无限MP群的逆系统的逆极限也是MP。最后，当G$ G$是有限生成时，我们建立了G$ G$实际上必须是有限的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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