# There exist subgroups of arbitrarily large subnormal depth

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This is a statement of the form: there exist subnormal subgroups of arbitrarily large subnormal depth satisfying certain conditions.

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## Statement

Let be a positive integer. Then, we can find a group and a subgroup such that is a -subnormal subgroup of but is *not* a -subnormal subgroup of . In other words, the subnormal depth of in is precisely . Equivalently, there exists a series of subgroups:

with each normal in , and there exists *no* series of length with the property.

## Related facts

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

## Proof

### Example of the dihedral group

`Further information: dihedral group`

Let be the dihedral group of order . Specifically, we have:

.

Let be the two-element subgroup generated by :

.

- is -subnormal in . Consider the series:

.

Each subgroup has index two in its predecessor, and is thus normal. The series has length , so is -subnormal in .

- is not -subnormal in : To see this, note that the above subnormal series is a subnormal series of minimum length, because, starting from the right, each subgroup is the normal closure of in the group to its right.