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arXiv · 2611.03366

Finite groups with all involutions fused

Zheng Guo·Yan Zhou Zhu
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Abstract

We characterize finite groups whose involutions form a single fusion class. In the solvable case, we show that the $2$-length is $1$, extending Thompson's theorem which states that a solvable group with all involutions conjugate has $2$-length $1$. For a nonsolvable group $G$, we prove that if its involutions are fused then they are conjugate; moreover, $G/\mathbf{O}_{2'}(G)$ is almost quasisimple. As an application, we obtain a restriction on $m$ for which an $m$-pyramidal Kirkman triple system can be realized under a nonsolvable group.

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Zheng Guo, Yan Zhou Zhu. 2026-11-03. Finite groups with all involutions fused. https://arxiv.org/abs/2611.03366

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