# Descendant not implies 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., descendant subgroup) neednotsatisfy the second subgroup property (i.e., subnormal subgroup)

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

Get more facts about descendant subgroup|Get more facts about subnormal subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property descendant subgroup but not subnormal subgroup|View examples of subgroups satisfying property descendant subgroup and subnormal subgroup

## Contents

## Statement

A descendant subgroup of a group need not be subnormal.

## Related facts

- Normality is not transitive
- There exist subgroups of arbitrarily large subnormal depth
- There exist subgroups of arbitrarily large descendant depth
- Ascendant not implies subnormal

## Definitions used

### Descendant subgroup

`Further information: Descendant subgroup`

### Subnormal subgroup

`Further information: Subnormal subgroup`

## Related facts

## Proof

### Example of the group of 2-adic integers

Let be the group of 2-adic integers under addition. This is the inverse limit of the chain:

.

Let be the semidirect product of with a group of order two, acting via the inverse map.

- is a descendant subgroup of : Consider a descending chain defined as follows. , and is the kernel of the quotient map to . Define as the semidirect product of with . Then, the form a descending chain of subgroups, each having index two in its predecessor, so each is normal in its predecessor. The intersection of all the s is equal to , and thus, is a descendant subgroup of .
- is not a subnormal subgroup of : If were a -subnormal subgroup of , then the image of in would be a -subnormal subgroup of for every . On the other hand, we know that the image of in has subnormal depth exactly in , which is a contradiction for . (For more on this, refer the example of the dihedral group in: there exist subgroups of arbitrarily large subnormal depth).

### Example of the infinite dihedral group

Suppose is the infinite dihedral group with cyclic maximal subgroup (isomorphic to the group of integers) and a complementary subgroup of order two. Then, is a descendant subgroup, as it is the intersection of the descending chain:

Note that each member has index two in its predecessor, hence is normal in its predecessor.

This example is very similar in nature to the 2-adic example.