# Odd-order class two group

From Groupprops

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: odd-order group and group of nilpotency class two

View other group property conjunctions OR view all group properties

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and Baer Lie group

View other group property conjunctions OR view all group properties

## Contents

## Definition

An **odd-order class two group** is defined in the following equivalent ways:

- It is an odd-order group that is also a nilpotent group of class at most two.
- It is a finite group that is also a Baer Lie group.
- It is a finite nilpotent group, hence an internal direct product of its Sylow subgroups, each of which has class at most two.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

odd-order abelian group | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

odd-order group | |FULL LIST, MORE INFO | |||

odd-order nilpotent group | |FULL LIST, MORE INFO | |||

group of nilpotency class two | |FULL LIST, MORE INFO | |||

Baer Lie group | |FULL LIST, MORE INFO | |||

Lazard Lie group | (via Baer Lie group) | Baer Lie group|FULL LIST, MORE INFO |