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Non-inner automorphisms of order $p$ in finite $p$-groups admitting cyclic center

Xuesong Ma, Wei Xu

TL;DR

The paper addresses the existence of non-inner automorphisms of order $p$ in finite non-abelian $p$-groups with cyclic center under a centralizer condition. It develops a derivation- and cohomology-based framework, including CR-modules and $H^1$-type analyses, to construct automorphisms of order $p$ from nontrivial derivations. The main contribution is a specific sufficient condition, $C_G(Z( olimits olinebreak ( olimits olinebreak ) )) leq $, under which such automorphisms exist, extending prior results (e.g., Marian) and providing a cohesive method to produce order-$p$ automorphisms in broad classes of $p$-groups. This advances the understanding of Gaschütz-type questions by connecting central structure, derivations, and automorphism generation in a unified approach.

Abstract

Let $G$ be a finite non-abelian $p$-group admitting cyclic center and $p$ be an odd prime. In this paper, we prove that if $C_{G}(Z(γ_{3}(G)G^{p}))\nleqslantγ_{3}(G)G^{p}$, then $G$ has a non-inner automorphism of order $p$.

Non-inner automorphisms of order $p$ in finite $p$-groups admitting cyclic center

TL;DR

The paper addresses the existence of non-inner automorphisms of order in finite non-abelian -groups with cyclic center under a centralizer condition. It develops a derivation- and cohomology-based framework, including CR-modules and -type analyses, to construct automorphisms of order from nontrivial derivations. The main contribution is a specific sufficient condition, , under which such automorphisms exist, extending prior results (e.g., Marian) and providing a cohesive method to produce order- automorphisms in broad classes of -groups. This advances the understanding of Gaschütz-type questions by connecting central structure, derivations, and automorphism generation in a unified approach.

Abstract

Let be a finite non-abelian -group admitting cyclic center and be an odd prime. In this paper, we prove that if , then has a non-inner automorphism of order .
Paper Structure (5 sections, 25 theorems, 50 equations)

This paper contains 5 sections, 25 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminary facts and results
  3. Existence of derivations
  4. Automorphisms with order $p$
  5. Main result and its proof

Key Result

Lemma 2.2

Let where $\psi(g)=g\delta(g)$ for all $g\in G$. Then $\mathcal{M}$ is a bijiective map.

Theorems & Definitions (31)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5: Lemma $2.8$ rus
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Definition 3.1
  • Lemma 3.2
  • ...and 21 more