# No nontrivial homomorphism to quotient group is not transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup having no nontrivial homomorphism to its quotient group)notsatisfying a subgroup metaproperty (i.e., transitive subgroup property).

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## Statement

It is possible to have a finite group and normal subgroups such that there is no nontrivial homomorphism from to and no nontrivial homomorphism from to , but there is a nontrivial homomorphism from to .

## Related facts

- Homomorph-containment is not transitive
- Full invariance is transitive
- No common composition factor with quotient group is transitive

## Proof

Let be the direct product of the symmetric group of degree three and a cyclic group of order three. Let be the first direct factor (i.e., the symmetric group of degree three) and be the unique subgroup of order three inside .

Then, there is no nontrivial homomorphism from to and none from to , but there is a nontrivial homomorphism from to , namely, an isomorphic mapping from to the second direct factor of .