Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characterizing finite solvable groups through the nilpotency probability

Andrea Lucchini

Abstract

Given a finite group $G$, we denote by $ν(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $ν(G)>1/12,$ then $G$ is solvable.

Characterizing finite solvable groups through the nilpotency probability

Abstract

Given a finite group , we denote by the probability that two randomly chosen elements of generate a nilpotent subgroup. We prove that if then is solvable.

Paper Structure

This paper contains 3 sections, 4 theorems, 31 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Details on the computation of $\tilde{\nu}(S)$

Key Result

Theorem 1

Let $G$ be a finite group. If $\nu(G)>\frac{1}{12},$ then $G$ is solvable. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (10)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • proof : Proof of Theorem \ref{['main']}
  • Lemma 6
  • proof