Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniformly semi-rational simple groups

Marco Vergani

Abstract

A finite group $G$ is called uniformly semi-rational if there exists an integer $r$ such that the generators of every cyclic sugroup $\langle x \rangle$ of $G$ lie in at most two conjugacy classes, namely $x^G$ or $(x^r)^G$. In this paper, we provide a classification of uniformly semi-rational non-abelian simple groups with particular focus on alternating groups.

Uniformly semi-rational simple groups

Abstract

A finite group is called uniformly semi-rational if there exists an integer such that the generators of every cyclic sugroup of lie in at most two conjugacy classes, namely or . In this paper, we provide a classification of uniformly semi-rational non-abelian simple groups with particular focus on alternating groups.

Paper Structure

This paper contains 3 sections, 6 theorems, 18 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Uniformly semi-rational alternating groups
  3. Non-alternating simple groups

Key Result

Lemma 2.1

Let $\pi\in A_n$ and consider the conjugacy class $\pi^{S_n}$. Then $\pi^{S_n}$ splits into two conjugacy classes of $A_n$ if and only if the numbers in the partition $\alpha(\pi)$ are pairwise distinct and odd.

Theorems & Definitions (20)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Definition 2.7
  • ...and 10 more