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Finite groups with all involutions fused
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/O2'(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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BibTeX
@misc{guo2026finite,
title = {{Finite groups with all involutions fused}},
author = {Zheng Guo and Yan Zhou Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03366},
eprint = {2611.03366},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03366},
}
RIS
TY - UNPB
TI - Finite groups with all involutions fused
AU - Zheng Guo
AU - Yan Zhou Zhu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03366
UR - https://arxiv.org/abs/2611.03366
AB - 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/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.
ER -