# Characteristic direct factor of nilpotent group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: characteristic direct factor with a group property imposed on theambient group: nilpotent group

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Contents

## Definition

A subgroup of a group is termed a **characteristic direct factor of nilpotent group** if it satisfies the following equivalent conditions:

- is a nilpotent group and is a characteristic direct factor of (i.e., is both a characteristic subgroup of and a direct factor of ).
- is a nilpotent group and is a fully invariant direct factor of (i.e., is both a fully invariant subgroup of and a direct factor of ). This has other equivalent formulations; see equivalence of definitions of fully invariant direct factor.

### Equivalence of definitions

`Further information: equivalence of definitions of characteristic direct factor of nilpotent group`

The equivalence follows indirectly from the fact that nontrivial subgroup of nilpotent group has nontrivial homomorphism to center.

## Relation with other properties

### Stronger properties

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

characteristic direct factor of abelian group | |FULL LIST, MORE INFO |