PreprintCC BY-NC-ND 4.0arXiv
Involutory automorphisms of finite groups of odd order
Abstract
For a finite group G we write γ∞(G) to denote the nilpotent residual of G, that is, the intersection of all terms of the lower central series. The following theorem is proved: Let G be a finite group of odd order admitting an involutory automorphism φ such that G=[G,φ] and suppose that γ∞(CG(φ)) has order m. Then the order of γ∞(G') is bounded by a function depending only on m. This complements several earlier results on groups of odd order admitting involutory automorphisms.
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BibTeX
@misc{shumyatsky2026involutory,
title = {{Involutory automorphisms of finite groups of odd order}},
author = {Pavel Shumyatsky and Gabriella Cristina de Souza},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09456},
eprint = {2609.09456},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09456},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Involutory automorphisms of finite groups of odd order
AU - Pavel Shumyatsky
AU - Gabriella Cristina de Souza
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09456
UR - https://arxiv.org/abs/2609.09456
AB - For a finite group G we write γ_∞(G) to denote the nilpotent residual of G, that is, the intersection of all terms of the lower central series. The following theorem is proved: Let G be a finite group of odd order admitting an involutory automorphism φ such that G=[G,φ] and suppose that γ_∞(C_G(φ)) has order m. Then the order of γ_∞(G') is bounded by a function depending only on m. This complements several earlier results on groups of odd order admitting involutory automorphisms.
ER -