# Normal not implies right-transitively fixed-depth subnormal

From Groupprops

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., right-transitively fixed-depth subnormal subgroup)

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

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

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

## Statement

There exists a group and a normal subgroup of such that is *not* right-transitively fixed-depth subnormal in . In other words, for any , there exists a -subnormal subgroup of such that is *not* -subnormal in .

## Related facts

- Normality is not transitive
- There exist subgroups of arbitrarily large subnormal depth
- Descendant not implies subnormal, Ascendant not implies subnormal
- Normal not implies left-transitively fixed-depth subnormal
- Characteristic not implies right-transitively fixed-depth subnormal

## Proof

### Example of the infinite dihedral group

Let be the infinite dihedral group, i.e., the group given by:

.

Let . is a subgroup of index two in , and is normal. For any , consider the subgroup:

.

Then:

- The subnormal depth of in is . In particular, is -subnormal in .
- The subnormal depth of in is . In particular, is not -subnormal in .