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 |
