# P-elementary group

From Groupprops

Revision as of 21:17, 1 March 2009 by Vipul (talk | contribs) (Elementary group moved to P-elementary group)

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter

View other prime-parametrized group properties | View other group properties

## Contents

## Definition

### Symbol-free definition

Let be a prime. A finite group is termed -elementary if it is the direct product of a -group and a cyclic group of order relatively prime to .

A group is termed elementary if it is -elementary for some prime .

### Definition with symbols

Let be a prime. A finite group is termed -elementary if is the internal direct product of a -subgroup and a cyclic subgroup whose order is relatively prime to .

A group is termed elementary if it is -elementary for some prime .

## Property theory

### Relation with other properties

Elementary groups are nilpotent. This is because cyclic groups are nilpotent, and -groups are also nilpotent, and a product of nilpotent groups is nilpotent.