Equivalence of definitions of intermediately characteristic subgroup of finite abelian group

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Statement

The following are equivalent for a subgroup H of a finite abelian group G:

  1. H is an Intermediately characteristic subgroup (?) of G, i.e., H is a characteristic subgroup in every subgroup of G containing it.
  2. H is an Intermediately fully invariant subgroup (?) of G, i.e., H is a fully invariant subgroup in every subgroup of G containing it.
  3. H is an Isomorph-containing subgroup (?) of G -- it contains every subgroup of G isomorphic to it. Equivalently, H is an Isomorph-free subgroup (?) of G.
  4. H is a Homomorph-containing subgroup (?) of G -- it contains every subgroup of G that is a homomorphic image of it.
  5. For every prime p, the p-Sylow subgroup of H is an omega subgroup of the corresponding p-Sylow subgroup of G.

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