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Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster
About this Title
Alexander Bors, Michael Giudici and Cheryl E. Praeger
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 287, Number 1427
ISBNs: 978-1-4704-6544-5 (print); 978-1-4704-7543-7 (online)
DOI: https://doi.org/10.1090/memo/1427
Published electronically: June 28, 2023
Keywords: Finite groups,
Automorphism orbits,
Element orders,
Simple groups,
Monster simple group
Table of Contents
Chapters
- 1. Introduction
- 2. Notation
- 3. Proof of Theorem 1.1.3
- 4. Proof of Theorem 1.1.2(1)
- 5. Proof of Theorem 1.1.2(2)
Abstract
For a finite group $G$, we denote by $\omega (G)$ the number of $Aut(G)$-orbits on $G$, and by $o(G)$ the number of distinct element orders in $G$. In this paper, we are primarily concerned with the two quantities $\mathfrak {d}(G)≔\omega (G)-o(G)$ and $\mathfrak {q}(G)≔\omega (G)/o(G)$, each of which may be viewed as a measure for how far $G$ is from being an AT-group in the sense of Zhang (that is, a group with $\omega (G)=o(G)$). We show that the index $|G:Rad(G)|$ of the soluble radical $Rad(G)$ of $G$ can be bounded from above both by a function in $\mathfrak {d}(G)$ and by a function in $\mathfrak {q}(G)$ and $o(Rad(G))$. We also obtain a curious quantitative characterisation of the Fischer-Griess Monster group $M$.- R. Abbott, J. Bray, S. Linton, S. Nickerson, S. Norton, R. Parker, I. Suleiman, J. Tripp, P. Walsh, and R. Wilson, ATLAS of Finite Group Representations – Version 3, http://brauer.maths.qmul.ac.uk/Atlas/v3/, online resource.
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