Equivalence of definitions of fully invariant direct factor

This article gives a proof/explanation of the equivalence of multiple definitions for the term fully invariant direct factor
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a Direct factor (?) $H$ of a group $G$:

1. $H$ is a Fully invariant subgroup (?) of $G$, i.e., every endomorphism of $G$ sends $H$ to itself. In other words, $H$ is a fully invariant direct factor.
2. $H$ is a Homomorph-containing subgroup (?) of $G$, i.e., for any homomorphism of groups from $H$ to $G$, the image of $H$ is contained in $H$.
3. $H$ is an Isomorph-containing subgroup (?) of $G$, i.e., $H$ contains any subgroup of $G$ isomorphic to $H$.

Proof

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