# Complete and potentially fully invariant implies homomorph-containing

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## Contents

## Statement

Suppose is a subgroup of such that is a complete group. Suppose further that is a potentially fully invariant subgroup of , i.e., there is a group containing such that is a fully invariant subgroup of . Then, is a homomorph-containing subgroup of : it contains every homomorphic image of itself in .

## Related facts

- Fully normalized and potentially fully invariant implies centralizer-annihilating endomomorphism-invariant
- Normal not implies potentially fully invariant

## Facts used

- Fully invariant implies normal
- Equivalence of definitions of complete direct factor: A complete subgroup is a normal subgroup if and only if it is a direct factor.
- Equivalence of definitions of fully invariant direct factor: For a direct factor, being fully invariant is equivalent to being a homomorph-containing subgroup.
- Homomorph-containment satisfies intermediate subgroup condition

## Proof

**Given**: A group , a complete subgroup . A group containing such that is fully invariant in .

**To prove**: is a homomorph-containing subgroup of .

**Proof**:

- is a normal subgroup of : This follows from fact (1).
- is a direct factor of : This follows from the previous step and fact (2).
- is a homomorph-containing subgroup of : This follows from the previous step and fact (3).
- is a homomorph-containing subgroup of : This follows from the previous step and fact (4).