No nontrivial homomorphism to quotient group not implies complemented normal

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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., normal subgroup having no nontrivial homomorphism to its quotient group) need not satisfy the second subgroup property (i.e., complemented normal subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about normal subgroup having no nontrivial homomorphism to its quotient group|Get more facts about complemented normal subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup having no nontrivial homomorphism to its quotient group but not complemented normal subgroup|View examples of subgroups satisfying property normal subgroup having no nontrivial homomorphism to its quotient group and complemented normal subgroup

Statement

It is possible to have a group G and a subgroup H such that H is a normal subgroup having no nontrivial homomorphism to its quotient group (i.e., the set \operatorname{Hom}(H,G/H) is a singleton set comprising the trivial homomorphism) but H is not a complemented normal subgroup in G.

Related facts

Proof

We take the following:

This example works because: