Equivalence of definitions of fully invariant direct factor

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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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