Normal not implies image-potentially fully invariant
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) need not satisfy the second subgroup property (i.e., image-potentially fully invariant subgroup)
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It is possible to have a normal subgroup of a group that is not an image-potentially fully invariant subgroup of , i.e., it is not possible to have a surjective homomorphism and a subgroup of such that .
- NIPC theorem: This states that every normal subgroup is image-potentially a characteristic subgroup.
- NPC theorem: This states that every normal subgroup is potentially a characteristic subgroup.
Suppose is the free group of rank two and is a normal subgroup of that is not a fully invariant subgroup of . In other words, there exists an endomorphism of <mah>G</math> such that is not contained inside .
Suppose is a surjective homomorphism and is a subgroup of such that . We show that is not fully invariant in .
Suppose is the kernel of . Since is a free group, is a complemented normal subgroup, so there exists a complement to in with an isomorphism such that if is the retraction with kernel and image , then . In particular, restricted to , .
Now, consider the endomorphism of defined as . Then, we see that . Thus, . But is not contained in , so is not contained in , so is not contained in . In particular, is not contained in . Hence, is not fully invariant in .