# Monolith is fully invariant in co-Hopfian group

From Groupprops

## Statement

If a Co-Hopfian group (?) (for instance, a Finite group (?)) has a Monolith (?) (a minimal normal subgroup contained in every nontrivial normal subgroup) then the monolith is a fully characteristic subgroup.

## Related facts

### Applications

## Facts used

## Proof

**Given**: A co-Hopfian group , a minimal normal subgroup contained in every nontrivial normal subgroup of , an endomorphism of .

**To prove**: .

**Proof**: If is not injective, it has a kernel. The kernel is a nontrivial normal subgroup, so it contains , so is trivial, and hence .

If is injective, then its image is a subgroup of isomorphic to . Since we assumed to be co-Hopfian, , so is surjective. But then, by fact (1), is nontrivial and normal, so , so , completing the proof.