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On the non-inner automorphism conjecture of finite $p$-groups

Sandeep Singh, Hemant Kalra, Rohit Garg

Abstract

A long-standing conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we settle the conjecture for a finite $p$-group ($p >2$) of nilpotency class $n$ with certain conditions.

On the non-inner automorphism conjecture of finite $p$-groups

Abstract

A long-standing conjecture asserts that every finite non-abelian -group has a non-inner automorphism of order . In this paper, we settle the conjecture for a finite -group () of nilpotency class with certain conditions.

Paper Structure

This paper contains 4 sections, 4 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Acknowledgement
  4. Conflict of Interest

Key Result

Theorem 2.1

Let $G$ be a finite $p$-group ($p>2$) of class $n$ such that $|\gamma_{n}(G)|=\hbox{exp}(\gamma_{n-1}(G))=p$ and $Z(C_G(x)) \le \gamma_{n-1}(G)$ for all $x \in \gamma_{n-1}(G) \setminus Z(G)$. Then $G$ has a non-inner automorphism of order $p$ that fixes $\Phi(G)$ element-wise.

Theorems & Definitions (8)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Example 2.5