# Homomorph-containing not implies no nontrivial homomorphism to quotient group

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., homomorph-containing subgroup) neednotsatisfy the second subgroup property (i.e., normal subgroup having no nontrivial homomorphism to its quotient group)

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## Statement

It is possible to have a group and a subgroup such that:

- is a homomorph-containing subgroup in , i.e., for any homomorphism in the image of is contained in .
- is
*not*a normal subgroup having no nontrivial homomorphism to its quotient group. In other words, there exists a nontrivial homomorphism from to .

## Proof

Take the following:

- is cyclic group:Z4.
- is the subgroup Z2 in Z4 (hence isomorphic to cyclic group:Z2), which is the unique subgroup generated by elements of order two.

Then:

- is homomorph-containing in : This is because the image of under any homomorphism must also have exponent at most two, hence must be inside .
- There is a nontrivial homomorphism from to : In fact, both are isomorphic to cyclic group:Z2.