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The Product of a Generalized Quaternion Group And a Cyclic Group

Shaofei Du, Hao Yu, Wenjuan Luo

TL;DR

The paper characterizes skew product groups $X=GC$ with $G\in\{Q,D\}$ and a cyclic $C$ with $G\cap C=1$, and obtains a complete classification for $X(Q)$ when $C$ is core-free, while also presenting a self-contained classification for $X(D)$ using a distinct approach. It reduces the problem to the analysis of a maximal subgroup $M$ containing $\langle c\rangle$, the core $M_X$, and the quotient $X/M_X$, showing that the latter can only be among $\mathbb{Z}_2$, $D_8$, $A_4$, or $S_4$, and then deriving explicit presentations for $X$ in each case via stepwise cyclic extensions. The proofs combine classical group theory—solvable groups, $p$-groups, permutation representations, and extension theory—with detailed casework to produce concrete generators, relations, and parameter congruences, including additional core-free constraints such as $\langle c\rangle_X=1$ in several instances. These results connect to skew morphisms and regular Cayley maps on $Q$ and $D$, offering a structural framework for understanding automorphism actions and factorizations in these classical groups. Overall, the work advances the theory of exact factorizations and skew product constructions by delivering explicit classifications and presentations that can be used in computational and theoretical contexts.

Abstract

Let $X(Q)=QC$ be a group, where $Q$ is a generalized quaternion group and $C$ is a cyclic group such that $Q\cap C=1$. In this paper, $X(Q)$ will be characterized and moreover, a complete classification for that will be given, provided $C$ is core-free. For the reason of self-constraint, in this paper a classification of the group $X(D)=DC$ is also given, where $D$ is a dihedral group and $C$ is a cyclic group such that $D\cap C=1$ and $C$ is core-free. Remind that the group $X(D)$ was recently classified in [12], based on a number of papers on skew-morphisms of dihedral groups. In this paper, a different approach from that in [12] will be used.

The Product of a Generalized Quaternion Group And a Cyclic Group

TL;DR

The paper characterizes skew product groups with and a cyclic with , and obtains a complete classification for when is core-free, while also presenting a self-contained classification for using a distinct approach. It reduces the problem to the analysis of a maximal subgroup containing , the core , and the quotient , showing that the latter can only be among , , , or , and then deriving explicit presentations for in each case via stepwise cyclic extensions. The proofs combine classical group theory—solvable groups, -groups, permutation representations, and extension theory—with detailed casework to produce concrete generators, relations, and parameter congruences, including additional core-free constraints such as in several instances. These results connect to skew morphisms and regular Cayley maps on and , offering a structural framework for understanding automorphism actions and factorizations in these classical groups. Overall, the work advances the theory of exact factorizations and skew product constructions by delivering explicit classifications and presentations that can be used in computational and theoretical contexts.

Abstract

Let be a group, where is a generalized quaternion group and is a cyclic group such that . In this paper, will be characterized and moreover, a complete classification for that will be given, provided is core-free. For the reason of self-constraint, in this paper a classification of the group is also given, where is a dihedral group and is a cyclic group such that and is core-free. Remind that the group was recently classified in [12], based on a number of papers on skew-morphisms of dihedral groups. In this paper, a different approach from that in [12] will be used.
Paper Structure (18 sections, 40 theorems, 256 equations, 1 table)

This paper contains 18 sections, 40 theorems, 256 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Elementary facts
  5. Proof of Theorem
  6. Proof of Theorem
  7. Proof of Theorem
  8. $M=\langle a\rangle \langle c\rangle$
  9. $M=\langle a^2\rangle\langle c\rangle$ and $X/M_X\cong D_8$
  10. $M=\langle a^2\rangle\langle c\rangle$ and $X/M_X\cong A_4$
  11. $M=\langle a^4\rangle\langle c\rangle$ and $X/M_X\cong S_4$
  12. $M=\langle a^3\rangle\langle c\rangle$ and $X/M_X\cong S_4$
  13. Proof of Theorem
  14. $M=\langle a\rangle \langle c\rangle$
  15. $M=\langle a^2\rangle\langle c\rangle$ and $X/M_X\cong D_8$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $G\in\{ Q,\,D\}$ and $X=G\langle c\rangle\in \{X(Q),\,X(D)\}$, where $\hbox{\rm o}(c)=m\ge 2$ and $G\cap \langle c\rangle=1$. Let $M$ be the subgroup of the biggest order in $X$ such that $\langle c\rangle \le M\subseteqq \langle a\rangle \langle c\rangle$. Then one of items in Tables holds.

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 2.1
  • Proposition 2.2
  • ...and 56 more