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The influence of the nilpotentlizers on group structur

N. Ahmadkhah, M. Zarrin

Abstract

For a finite group $G$ and an element $x\in G$, the subset $$ nil_G(x)=\{y\in G \mid <x,y>~~ is ~~ nilpotent\}$$ is called nilpotentizer of $x$ in $G$. In this paper, we give two solvabilty criteria for a finite group by the structure and the size of nilpotentizer of an element on finite group. In fact, we show that if there exists an element $x$ of $G$ such that $nil_G(x)$ generates a maximal subgroup of $G$ and the simple commutator of weight $2 ~~or ~~3$ of elements of $nil_G(x)$ is equal to $1$ or $|nil_G(x)|= p^n$, where $p$ is prime and $n=1, 2$. Then $G$ is a solvable group.

The influence of the nilpotentlizers on group structur

Abstract

For a finite group and an element , the subset is called nilpotentizer of in . In this paper, we give two solvabilty criteria for a finite group by the structure and the size of nilpotentizer of an element on finite group. In fact, we show that if there exists an element of such that generates a maximal subgroup of and the simple commutator of weight of elements of is equal to or , where is prime and . Then is a solvable group.
Paper Structure (2 sections, 16 theorems, 4 equations)

This paper contains 2 sections, 16 theorems, 4 equations.

Table of Contents

  1. Introduction and Basic results
  2. Proofs

Key Result

Theorem 1.1

(see AZ) Let $G$ be a group and $x\in G$. Then $<x>\subseteq(<x>, Z(G))\subseteq C_G(x)\subseteq nil_G(x)$. $nil_G(x)$ is the union of all maximal nilpotent subgroups of $G$ containing $x$. $|nil_G(x)|$ is divisible by $|x|$. If $N$ is a normal subgroup of $G$ and $x,y\in G$, then A. $\frac{nil_G(x)

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Corollary 2.5
  • ...and 19 more