Subhomomorph-containment is transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., subhomomorph-containing subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Suppose . Suppose is a subhomomorph-containing subgroup of and is a subhomomorph-containing group of . Then, is a subhomomorph-containing subgroup of .
- Homomorph-containment is not transitive
- Subhomomorph-containing implies right-transitively homomorph-containing
- Full invariance is transitive
Given: Groups . A homomorphism for a subgroup of .
To prove: is a subgroup of .
Proof: Since is a subgroup of , is a subgroup of . Thus, is a homomorphism from a subgroup of . Since is subhomomorph-containing in , is contained in . Thus, can be viewed as a map from to .
Thus, is a map from a subgroup of . Since is subhomomorph-containing in , , and we are done.