# Normal not implies left-transitively fixed-depth subnormal

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., left-transitively fixed-depth subnormal subgroup)

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Get more facts about normal subgroup|Get more facts about left-transitively fixed-depth subnormal subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not left-transitively fixed-depth subnormal subgroup|View examples of subgroups satisfying property normal subgroup and left-transitively fixed-depth subnormal subgroup

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., subnormal subgroup) neednotsatisfy the second subgroup property (i.e., left-transitively fixed-depth subnormal subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about subnormal subgroup|Get more facts about left-transitively fixed-depth subnormal subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property subnormal subgroup but not left-transitively fixed-depth subnormal subgroup|View examples of subgroups satisfying property subnormal subgroup and left-transitively fixed-depth subnormal subgroup

## Statement

It is possible to have a group and a normal subgroup such that for every , there exists a group containing as a -subnormal subgroup, but in which is *not* a -subnormal subgroup.

Note that this also gives an example where is a subnormal subgroup of , and for every , there exists a group containing as a -subnormal subgroup, but in which is *not* a -subnormal subgroup.

## Related facts

- Left transiter of normal is characteristic
- Cofactorial automorphism-invariant implies left-transitively 2-subnormal
- Normality is not transitive
- There exist subgroups of arbitrarily large subnormal depth
- Ascendant not implies subnormal, descendant not implies subnormal
- Normal not implies right-transitively fixed-depth subnormal

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8`

Let be the dihedral group of order eight, given by:

.

Let be the subgroup of generated by and . has index two in and is hence normal. (subgroup of index two is normal).

For any define as the group:

.

In other words, is a dihedral group of order . Consider as a subgroup of by identifying and . Then:

- The subnormal depth of in is .
- The subnormal depth of in is .

This completes the proof.