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Intense Automorphisms of Finite Groups
About this Title
Mima Stanojkovski
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 273, Number 1341
ISBNs: 978-1-4704-5003-8 (print); 978-1-4704-6811-8 (online)
DOI: https://doi.org/10.1090/memo/1341
Published electronically: November 3, 2021
Keywords: Classification,
finite $p$-groups,
intense automorphisms,
intensity
Table of Contents
Chapters
- List of Symbols
- 1. Introduction
- 2. Coprime Actions
- 3. Intense Automorphisms
- 4. Intensity of Groups of Class 2
- 5. Intensity of Groups of Class 3
- 6. Some Structural Restrictions
- 7. Higher Nilpotency Classes
- 8. A Disparity between the Primes
- 9. The Special Case of 3-groups
- 10. Obelisks
- 11. The Most Intense Chapter
- 12. High Class Intensity
- 13. Intense Automorphisms of Profinite Groups
Abstract
Let $G$ be a group. An automorphism of $G$ is called intense if it sends each subgroup of $G$ to a conjugate; the collection of such automorphisms is denoted by $\operatorname {Int}(G)$. In the special case in which $p$ is a prime number and $G$ is a finite $p$-group, one can show that $\operatorname {Int}(G)$ is the semidirect product of a normal $p$-Sylow and a cyclic subgroup of order dividing $p-1$. In this paper we classify the finite $p$-groups whose groups of intense automorphisms are not themselves $p$-groups. It emerges from our investigation that the structure of such groups is almost completely determined by their nilpotency class: for $p>3$, they share a quotient, growing with their class, with a uniquely determined infinite 2-generated pro-$p$ group.- H. A. Bender, A determination of the groups of order $p^5$, Ann. of Math. (2) 29 (1927/28), no. 1-4, 61–72. MR 1502819, DOI 10.2307/1967981
- Norman Blackburn, Generalizations of certain elementary theorems on $p$-groups, Proc. London Math. Soc. (3) 11 (1961), 1–22. MR 122876, DOI 10.1112/plms/s3-11.1.1
- Henri Cohen, Number theory. Vol. I. Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, Springer, New York, 2007. MR 2312337
- J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$-groups, London Mathematical Society Lecture Note Series, vol. 157, Cambridge University Press, Cambridge, 1991. MR 1152800
- Jon González-Sánchez and Benjamin Klopsch, Analytic pro-$p$ groups of small dimensions, J. Group Theory 12 (2009), no. 5, 711–734. MR 2554763, DOI 10.1515/JGT.2009.006
- Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528, DOI 10.1017/CBO9780511607219
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
- I. Martin Isaacs, Finite group theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, Providence, RI, 2008. MR 2426855, DOI 10.1090/gsm/092
- G. Klaas, C. R. Leedham-Green, and W. Plesken, Linear pro-$p$-groups of finite width, Lecture Notes in Mathematics, vol. 1674, Springer-Verlag, Berlin, 1997. MR 1483894, DOI 10.1007/BFb0094086
- Thomas J. Laffey, The minimum number of generators of a finite $p$-group, Bull. London Math. Soc. 5 (1973), 288–290. MR 325763, DOI 10.1112/blms/5.3.288
- Serge Lang and John Tate, Principal homogeneous spaces over abelian varieties, Amer. J. Math. 80 (1958), 659–684. MR 106226, DOI 10.2307/2372778
- Luis Ribes and Pavel Zalesskii, Profinite groups, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 40, Springer-Verlag, Berlin, 2010. MR 2599132, DOI 10.1007/978-3-642-01642-4
- Mima Stanojkovski, Evolving groups, Arch. Math. (Basel) 111 (2018), no. 1, 3–12. MR 3816972, DOI 10.1007/s00013-018-1178-9