# Minimal normal implies characteristically simple

From Groupprops

Template:Subgroup-to-group property implicationThis article describes a fact or result that is not basic but it still well-established and standard. The fact may involve terms that are themselves non-basic

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This fact is an application of the following pivotal fact/result/idea:characteristic of normal implies normal

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## Contents

## Statement

### Verbal statement

Any group that occurs as a minimal normal subgroup of some group is characteristically simple.

## Definitions

### Minimal normal subgroup

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### Characteristically simple group

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## Related facts

### Similar facts for specific kinds of groups

- Minimal normal implies additive group of a field in solvable
- Minimal normal implies elementary Abelian in finite solvable
- Minimal normal implies contained in Omega-1 of center for nilpotent p-group
- Minimal normal implies central in nilpotent

### Converse

- Characteristically simple equals left-realization of minimal normal: Every characteristically simple group can be embedded as a minimal normal subgroup inside some group (in fact, inside its holomorph).

## Facts used

- Characteristic of normal implies normal: Every characteristic subgroup of a normal subgroup is normal in the whole group.

## Proof

### Hands-on proof

**Given**: is a minimal normal subgroup of a group .

**To prove**: is characteristically simple.

**Proof**: Suppose is a characteristic subgroup of . Then, by fact (1), is normal in . But since was *minimal* normal, either or is trivial. Thus, every characteristic subgroup of is either the whole of or is trivial.