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Groups with a conjugacy class that is the difference of two normal subgroups

Mark L. Lewis, Lucia Morotti, Emanuele Pacifici, Lucia Sanus, Hung P. Tong-Viet

Abstract

We consider finite groups having a conjugacy class that is the difference of two normal subgroups. That is, suppose $G$ is a group and $M$ and $N$ are normal subgroups so that $N < M$, and suppose that there is an element $g \in G$ so that the conjugacy class of $g$ is $M \setminus N$. We find a character-theoretic characterization of this condition, and we determine some structural properties of groups with such a conjugacy class. If we add the condition that $M/N$ is the unique minimal normal subgroup of $G/N$, then we obtain a generalization of a result by S.M. Gagola.

Groups with a conjugacy class that is the difference of two normal subgroups

Abstract

We consider finite groups having a conjugacy class that is the difference of two normal subgroups. That is, suppose is a group and and are normal subgroups so that , and suppose that there is an element so that the conjugacy class of is . We find a character-theoretic characterization of this condition, and we determine some structural properties of groups with such a conjugacy class. If we add the condition that is the unique minimal normal subgroup of , then we obtain a generalization of a result by S.M. Gagola.

Paper Structure

This paper contains 9 sections, 41 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Conjugacy classes that are the differences of two normal subgroups
  4. Proofs of Theorems \ref{['th:character']} and \ref{['th:class']}
  5. Central extensions of Gagola groups: $p$-groups
  6. Central extensions Gagola groups: Frobenius groups
  7. nonsolvable groups: a proof of Theorem \ref{['th:simple']} and Theorem \ref{['th:almostsemisimple']}
  8. Solvable groups: proofs of Theorems \ref{['th:hn_pgroup']}, \ref{['th:abelian-by-metanilpotent']}, and \ref{['th:Fittingheight']}
  9. Final remarks

Key Result

Theorem 1

Let $G$ be a group, let $N < M$ be normal subgroups of $G$, and let $g \in M \setminus N$. Then $g^G = M \setminus N$ if and only if every character in ${\rm Irr}(G/N)$ is constant on $M/N \setminus \{ N \}$ and every character in ${\rm Irr}(G \mid N)$ vanishes on $g$.

Theorems & Definitions (77)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 67 more