Exponential and weakly exponential subgroups of finite groups

Document Type : Research Paper

Authors

1 Department of Mathematics, William & Mary, Williamsburg, VA 23187, USA

2 Department of Mathematics, Compueter and Information Science, SUNY at Old Westbury, Old Westbury, NY 11568, USA

Abstract

Sabatini [L. Sabatini, Products of subgroups, subnormality, and relative orders of elements, Ars Math. Contemp., 24 no. 1 (2024) 9 pp.] defined a subgroup $H$ of $G$ to be an exponential subgroup if $x^{|G:H|} \in H$ for all $x \in G$, in which case we write H ≤exp G. Exponential subgroups are a generalization of normal (and subnormal) subgroups: all subnormal subgroups are exponential, but not conversely. Sabatini proved that all subgroups of a finite group $G$ are exponential if and only if $G$ is nilpotent. The purpose of this paper is to explore what the analogues of a simple group and a solvable group should be in relation to exponential subgroups. We say that an exponential subgroup H ≤exp G is exp-trivial if either $H = G$ or the exponent of $G$, $\exp(G)$, divides $|G:H|$, and we say that a group $G$ is exp-trivial if all exponential subgroups of $G$ are exp-trivial. We classify finite exp-simple groups by proving $G$ is exp-simple if and only if $\exp(G) = \exp(G/N)$ for all proper normal subgroups $N$ of $G$, and we illustrate how the class of exp-simple groups differs from the class of simple groups. Furthermore, in an attempt to overcome the obstacle that prevents all subgroups of a generic solvable group from being exponential, we say that a subgroup $H$ of $G$ is weakly exponential if, for all $x \in G$, there exists $g \in G$ such that $x^{|G:H|} \in H^g$. If all subgroups of $G$ are weakly exponential, then $G$ is wexp-solvable. We prove that all solvable groups are wexp-solvable and almost all symmetric and alternating groups are not wexp-solvable. Finally, we completely classify the groups $PSL(2,q)$ that are wexp-solvable. We show that if $\pi(n)$ denotes the number of primes less than $n$ and $w(n)$ denotes the number of primes $p$ less than $n$ such that $PSL(2,p)$ is wexp-solvable, then
\[ \lim_{n \to \infty} \frac{w(n)}{\pi(n)} = \frac{1}{4}.\]

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Main Subjects

References

[1] L. E. Dickson, Linear groups: With an exposition of the Galois field theory, With an introduction by W. Magnus. Dover Publications, Inc., New York, 1958.
[2] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.12.2, 2022.
https://www.gap-system.org.
[3] M. Garonzi, L.-C. Kappe and E. Swartz, On integers that are covering numbers of groups, Exp. Math., 31 no. 2 (2022) 425–443.
[4] I. M. Isaacs, Finite group theory, Graduate Studies in Mathematics, 92, American Mathematical Society, Providence, RI, 2008.
[5] L. Sabatini, Products of subgroups, subnormality, and relative orders of elements, Ars Math. Contemp., 24 no. 1 (2024) 9 pp.
[6] N. J. Werner, A note on finite nilpotent groups. Amer. Math. Monthly, 131 no. 9 (2024) 803–805. https://doi.org/10.1080/00029890.2024.2379751
International Journal of Group Theory
Volume 14, Issue 4 - Serial Number 4
December 2025
Pages 263-283

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History

  • Receive Date: 07 August 2024
  • Revise Date: 25 October 2024
  • Accept Date: 31 October 2024
  • Published Online: 13 November 2024

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