Complete and potentially fully invariant implies homomorph-containing

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

Related facts

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

Facts used

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

Proof
Given: A group $$G$$, a complete subgroup $$H$$. A group $$K$$ containing $$G$$ such that $$H$$ is fully invariant in $$K$$.

To prove: $$H$$ is a homomorph-containing subgroup of $$G$$.

Proof:


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