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Complete and potentially fully invariant implies homomorph-containing

Contents

Statement

Suppose H is a subgroup of G such that H is a complete group. Suppose further that H is a potentially fully invariant subgroup of G, i.e., there is a group K containing G such that H is a fully invariant subgroup of K. Then, H is a homomorph-containing subgroup of G: it contains every homomorphic image of itself in G.

Related facts

Facts used

  1. Fully invariant implies normal
  2. Equivalence of definitions of complete direct factor: A complete subgroup is a normal subgroup if and only if it is a direct factor.
  3. Equivalence of definitions of fully invariant direct factor: For a direct factor, being fully invariant is equivalent to being a homomorph-containing subgroup.
  4. Homomorph-containment satisfies intermediate subgroup condition

Proof

Given: A group G, a complete subgroup H. A group K containing G such that H is fully invariant in K.

To prove: H is a homomorph-containing subgroup of G.

Proof:

  1. H is a normal subgroup of K: This follows from fact (1).
  2. H is a direct factor of K: This follows from the previous step and fact (2).
  3. H is a homomorph-containing subgroup of K: This follows from the previous step and fact (3).
  4. H is a homomorph-containing subgroup of G: This follows from the previous step and fact (4).