Transitive and intransitive subgroups of permutation groups

Demirhan, Arda; Miller, Jacob; Qiu, Yixu; Tucker, Thomas J.; Zhu, Zheng

doi:10.2140/involve.2025.18.547

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Transitive and intransitive subgroups of permutation groups

Arda Demirhan, Jacob Miller, Yixu Qiu, Thomas J. Tucker and Zheng Zhu

Vol. 18 (2025), No. 3, 547–553

DOI: 10.2140/involve.2025.18.547

Abstract

We treat the problem of finding transitive subgroups $G$ of $S_{n}$ containing normal subgroups $N_{1}$ and $N_{2}$ , with $N_{1}$ transitive and $N_{2}$ not transitive, such that $G ∕ N_{1} ≅ G ∕ N_{2}$ . We show that such $G$ exist whenever $n$ has a prime factor that also divides $ϕ (n)$ . We show that no such $G$ exist when $n = p q$ for $p < q$ with $p$ not dividing $q - 1$ .

Keywords

wreath products, transitive subgroups, permutation groups

Mathematical Subject Classification

Primary: 20B05

Secondary: 37P10

Milestones

Received: 20 October 2023

Revised: 20 December 2023

Accepted: 16 January 2024

Published: 28 April 2025

Communicated by Vadim Ponomarenko

Authors

Arda Demirhan
Department of Mathematics University of Rochester Rochester, NY United States

Jacob Miller
Department of Mathematics University of Iowa Iowa City, IA United States

Yixu Qiu
Courant Institute of Mathematical Sciences New York University New York, NY United States

Thomas J. Tucker
Department of Mathematics University of Rochester Rochester, NY United States

Zheng Zhu
Department of Mathematics University of Rochester Rochester, NY United States

Vol. 18, No. 3, 2025