On profinite groups with the Magnus Property

TL;DR

This paper investigates profinite groups with the Magnus Property, showing they are prosolvable, have limited prime divisors, and finitely generated groups are finite, revealing structural constraints of such groups.

Contribution

It proves that profinite MP groups are prosolvable, their quotients are MP, and finitely generated MP groups are finite, establishing new structural properties.

Findings

Profinite MP groups are prosolvable.

Prime divisors of MP groups are limited to 2, 3, 5, 7.

Finitely generated MP groups are finite.

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.