Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster
Seiten
2023
American Mathematical Society (Verlag)
978-1-4704-6544-5 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-6544-5 (ISBN)
For a finite group G, we denote by ?(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 d(G) := ?(G) ? o(G) and q(G) := ?(G)/ o(G).
For a finite group G, we denote by ?(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 d(G) := ?(G) ? o(G) and q(G) := ?(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 ?(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 d(G) and by a function in q(G) and o(Rad(G)). We also obtain a curious quantitative characterisation of the Fischer-Griess Monster group M.
For a finite group G, we denote by ?(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 d(G) := ?(G) ? o(G) and q(G) := ?(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 ?(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 d(G) and by a function in q(G) and o(Rad(G)). We also obtain a curious quantitative characterisation of the Fischer-Griess Monster group M.
Alexander Bors, Carleton University, Ottawa, Canada, The University of Western Australia, Crawley, Australia, and Radon Institute for Computational and Applied Mathematics, Linz, Austria. Michael Giudici, The University of Western Australia, Crawley, Australia. Cheryl E. Praeger, The University of Western Australia, Crawley, Australia.
| Erscheinungsdatum | 19.08.2023 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society ; Volume: 287 Number: 1427 |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| ISBN-10 | 1-4704-6544-2 / 1470465442 |
| ISBN-13 | 978-1-4704-6544-5 / 9781470465445 |
| Zustand | Neuware |
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