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arXiv · 2610.03151

Arithmetic properties of solubilizers

Luca Sabatini
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Abstract

Let $G$ be a finite group and $x \in G$. In 2013, Hai-Reuven proved that the cardinality of the centralizer of $x$ divides the cardinality of the solubilizer of $x$. Settling a conjecture of Mousavi, Poozesh and Zamani, we show that the cardinality of the normalizer of $\langle x \rangle$ is also a divisor. We actually give direct short proofs of more general results.

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Luca Sabatini. 2026-10-02. Arithmetic properties of solubilizers. https://arxiv.org/abs/2610.03151

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