# Normal not implies 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) neednotsatisfy the second subgroup property (i.e., potentially fully invariant subgroup)

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## Contents

## Statement

It is possible to have a normal subgroup of a group that is not a potentially fully invariant subgroup of -- in other words, there is no group containing such that is a fully invariant subgroup of .

## Related facts

- Normal not implies image-potentially fully invariant
- Normal not implies potentially verbal
- NPC theorem: Normal equals potentially
*characteristic*. - Normal equals potentially normal-subhomomorph-containing
- Complete and potentially fully invariant implies homomorph-containing
- Fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant

## Facts used

- Equivalence of definitions of complete direct factor: This states that for a complete subgroup, being normal is equivalent to being a direct factor.
- Equivalence of definitions of fully invariant direct factor: This states that for a direct factor, being a fully invariant subgroup is equivalent to being a homomorph-containing subgroup.
- Homomorph-containment satisfies intermediate subgroup condition

## Proof

### Example involving a complete group

`Further information: Complete and potentially fully invariant implies homomorph-containing`

Let be a nontrivial complete group. Define and . Clearly, is a normal subgroup of .

Suppose is a group containing , such that is fully invariant in . In particular, is normal in . Since is complete, it is a direct factor, so there exists a group that is a complement to , so as an internal direct product. Further, since is a subgroup of , has a subgroup, say , isomorphic to .

Then, consider the endomorphism of that sends to the trivial subgroup and isomorphically to the subgroup . This endomorphism does *not* send to within itself.

### More general example

`Further information: Fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant`

More generally, suppose is a fully normalized subgroup of that is normal in , but such that there is a homomorphism whose kernel contains such that is not contained in (in other words, is not a centralizer-annihilating endomorphism-invariant subgroup).
Then, is *not* a potentially fully invariant subgroup of .

Examples include:

- is the dihedral group of order 16, say , and . Then, and we have a homomorphism from to such that with kernel containing and with not contained in . Thus, is not a potentially fully invariant subgroup of .
`Further information: D8 is not potentially fully invariant in D16`