# Normal-homomorph-containing implies strictly characteristic

From Groupprops

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., normal-homomorph-containing subgroup) must also satisfy the second subgroup property (i.e., strictly characteristic subgroup)

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## Statement

### Statement with symbols

Suppose is a normal-homomorph-containing subgroup of a group : for any homomorphism such that is normal in , we have .

Then, is a strictly characteristic subgroup of : for any surjective endomorphism of , .

## Facts used

- Normality satisfies image condition: The image of a normal subgroup under a surjective homomorphism is a normal subgroup of the image.

## Proof

**Given**: A normal subgroup of a group such that whenever is a homomorphism such that is normal in , we have . A surjective endomorphism of .

**To prove**: .

**Proof**:

- (
**Given data used**: is normal in , is a surjective endomorphism): is a normal subgroup of : By fact (1), the image is a normal subgroup of . By surjectivity, we have , so is normal in . - (
**Given data used**: is normal-homomorph-containing): : Let be the restriction of to . Then by definition, and by step (1), is normal in . Since is normal-homomorph-containing, we get , so .