# Normal-isomorph-free not implies isomorph-free in finite

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-isomorph-free subgroup) neednotsatisfy the second subgroup property (i.e., isomorph-free subgroup)

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## Contents

## Statement

### Verbal statement

We may have a group with a normal subgroup such that there is no other *normal* subgroup isomorphic to it, but there are other non-normal subgroups isomorphic to it.

## Related facts

- Characteristic not implies characteristic-isomorph-free
- Characteristic-isomorph-free not implies normal-isomorph-free

## Proof

### An example involving the symmetric group

`Further information: symmetric group:S3`

Let be the symmetric group on three letters and be the cyclic group of order two.

- is a normal subgroup of . Further, if is cyclic of order two and normal in , then the projection of in is normal in . Since has no normal subgroups of order two, the projection of in is trivial, so . Thus, is a normal-isomorph-free subgroup of .
- On the other hand, is not isomorph-free in : We can find a two-element subgroup of , and we then have .

### An example involving the dihedral group

`Further information: dihedral group:D8`

Let be the dihedral group of order eight:

.

Let be the center of , so . Then:

- is normal-isomorph-free: It is the only normal subgroup of order two.
- is not isomorph-free: There are other subgroups of order two, such as .

### A generic example

For a group of prime power order, the following is true: if the center is of prime order, it is normal-isomorph-free. This follows from the fact that nilpotent implies center is normality-large: in a nilpotent group, the intersection between the center and any nontrivial normal subgroup is nontrivial. On the other hand, the center is rarely isomorph-free (an example where it *is* isomorph-free is the generalized quaternion group).

More generally, if is a -group and denotes the set of elements of order , then if is cyclic of order , it is normal-isomorph-free.