Homomorph-containment satisfies intermediate subgroup condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., homomorph-containing subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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A homomorph-containing subgroup of the whole group is also a homomorph-containing subgroup of any intermediate subgroup.
Statement with symbols
Suppose are groups, and is a homomorph-containing subgroup of . Then, is also a homomorph-containing subgroup of .
- Full invariance does not satisfy intermediate subgroup condition
- Homomorph-containing implies intermediately fully invariant
- Homomorph-containment does not satisfy image condition
Given: Groups such that is homomorph-containing in . A homomorphism .
To prove: is contained in .
Proof: Since , we can compose with the inclusion of in to get a homomorphism . Since is homomorph-containing in , , so .