Equivalence of definitions of image-closed characteristic subgroup of finite abelian group

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Statement

Suppose G in a finite abelian group and H is a subgroup of G. The following are equivalent:

  1. H is an Image-closed characteristic subgroup (?) of G: for any surjective homomorphism from G, the image of H is characteristic in the image of G.
  2. H is an Image-closed fully invariant subgroup (?) of G: for any surjective homomorphism from G, the image of H is fully invariant in the image of G.
  3. H is a Verbal subgroup (?) of G.
  4. For every prime p, the p-Sylow subgroup of H is an agemo subgroup of the p-Sylow subgroup of G.

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