# Self-centralizing and minimal normal implies strictly characteristic

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., self-centralizing minimal normal subgroup) must also satisfy the second subgroup property (i.e., strictly characteristic subgroup)

View all subgroup property implications | View all subgroup property non-implications

Get more facts about self-centralizing minimal normal subgroup|Get more facts about strictly characteristic subgroup

## Contents

## Statement

### Verbal statement

Any Minimal normal subgroup (?) that is also self-centralizing (i.e., contains its centralizer in the whole group) is strictly characteristic: it is invariant under any surjective endomorphism of the group.

## Definitions used

### Minimal normal subgroup

`Further information: minimal normal subgroup`

A minimal normal subgroup of a group is a nontrivial normal subgroup that does not properly contain any other nontrivial normal subgroup.

### Self-centralizing subgroup

`Further information: self-centralizing subgroup`

A subgroup of a group is termed **self-centralizing** if it contains its centralizer in the whole group.

### Strictly characteristic subgroup

`Further information: strictly characteristic subgroup`

A subgroup of a group is termed **strictly characteristic** if for every surjective endomorphism of , .

## Related facts

- Self-centralizing and minimal normal implies monolith
- Self-centralizing and minimal normal implies characteristic

## Facts used

- Self-centralizing and minimal normal implies monolith: In other words, a self-centralizing minimal normal subgroup is contained in every nontrivial normal subgroup.
- Monolith is strictly characteristic: A subgroup contained in every nontrivial normal subgroup is strictly characteristic.

## Proof

**Given**: A group , a minimal normal subgroup such that . A surjective endomorphism of .

**To prove**: .

**Proof**: Consider the subgroup . This is normal by fact (2), so by fact (1) either is trivial or . Since is surjective and is nontrivial, cannot be trivial. Thus, . This forces that , as desired.