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Class-preserving Coleman Automorphisms of Finite Groups with Semidihedral Sylow 2-Subgroups

Riccardo Aragona

Abstract

In this paper, we prove that finite groups with semidihedral Sylow 2-subgroup have Class-preserving Coleman outer automorphism group of odd order. As a consequence, these groups satisfy the normalizer problem. In particular, we extend some existing results in the literature concerning class-preserving Coleman automorphisms of finite groups with semidihedral Sylow 2-subgroups.

Class-preserving Coleman Automorphisms of Finite Groups with Semidihedral Sylow 2-Subgroups

Abstract

In this paper, we prove that finite groups with semidihedral Sylow 2-subgroup have Class-preserving Coleman outer automorphism group of odd order. As a consequence, these groups satisfy the normalizer problem. In particular, we extend some existing results in the literature concerning class-preserving Coleman automorphisms of finite groups with semidihedral Sylow 2-subgroups.

Paper Structure

This paper contains 3 sections, 14 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Proof of Theorem \ref{['main']}

Key Result

Theorem 2.1

If $G$ is a finite group with semidihedral Sylow 2-subgroup, then $G$ satisfies Property property; in other words, $\mathop{\mathrm{Out}}\nolimits_c(G)\cap \mathop{\mathrm{Out}}\nolimits_{Col}(G)$ is of odd order.

Theorems & Definitions (28)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Remark 1
  • Remark 2
  • Lemma 3.1
  • ...and 18 more