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Finite groups in which every proper characteristic subgroup is cyclic

Marco Damele, Fabio Mastrogiacomo

Abstract

Let $G$ be a finite non-cyclic, non-characteristically simple group with the property that all proper characteristic subgroups of $G$ are cyclic. We call such a group $\mathrm{CCS}$ group, short for \emph{Characteristic Cyclic}. In this paper, we provide a complete classification of these groups. As an application of our main result, we also make some progress toward the classification of minimal non-cyclic skew braces.

Finite groups in which every proper characteristic subgroup is cyclic

Abstract

Let be a finite non-cyclic, non-characteristically simple group with the property that all proper characteristic subgroups of are cyclic. We call such a group group, short for \emph{Characteristic Cyclic}. In this paper, we provide a complete classification of these groups. As an application of our main result, we also make some progress toward the classification of minimal non-cyclic skew braces.

Paper Structure

This paper contains 11 sections, 25 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Nilpotent $\mathrm{CCS}$ groups
  4. Non-nilpotent solvable $\mathrm{CCS}$ groups
  5. The case $P = C_p \times C_p$
  6. The case $P = C_{p^{\alpha}}$
  7. The case $P = D_{2^\alpha}$
  8. The case $P = Q_{2^\alpha}$
  9. Conclusion
  10. Perfect $\mathrm{CCS}$ groups
  11. Proofs of Theorems \ref{['Th2']} and \ref{['thm:skbr']}

Key Result

Theorem 1.1

Let $G$ be a $\mathrm{CCS}$ group. Then, one of the following holds. Conversely, each of these groups is a $\mathrm{CCS}$ group.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 31 more