# Monolith is 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., monolith) must also satisfy the second subgroup property (i.e., strictly characteristic subgroup)

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## Statement

### Verbal statement

If a group has a monolith (a Minimal normal subgroup (?) contained in every nontrivial normal subgroup), then that monolith is a strictly characteristic subgroup (it is invariant under any surjective endomorphism of the group).

## Related facts

## Facts used

- Normality satisfies inverse image condition: The inverse image of a normal subgroup under any homomorphism is normal.

## Proof

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

**To prove**: .

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