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arXiv · 2412.08470

On profinite groups with the Magnus Property

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 the only prime divisors of $|G|$ are $2$, $3$, $5$ and $7$, 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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Claude Marion, Pavel Zalesskii. 2024-12-11. On profinite groups with the Magnus Property. https://arxiv.org/abs/2412.08470

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