# Equivalence of definitions of fully invariant direct factor

From Groupprops

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 (?) of a group :

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

## Proof

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