# Complemented characteristic not implies left-transitively complemented normal

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., complemented characteristic subgroup) neednotsatisfy the second subgroup property (i.e., left-transitively complemented normal subgroup)

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

Get more facts about complemented characteristic subgroup|Get more facts about left-transitively complemented normal subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property complemented characteristic subgroup but not left-transitively complemented normal subgroup|View examples of subgroups satisfying property complemented characteristic subgroup and left-transitively complemented normal subgroup

## Contents

## Statement

### Statement with symbols

It is possible to have the following situation: groups such that is a complemented characteristic subgroup of and is a complemented normal subgroup of , but is not a complemented normal subgroup of .

## Related facts

### Opposite facts

### Converse

## Facts used

## Proof

### Example of the dihedral group of order sixteen

`Further information: Dihedral group:D16`

Let be a dihedral group of order . In other words, is given by the presentation:

.

Consider the subgroups:

.

- is complemented characteristic in : is a dihedral group of order eight, with as the unique cyclic normal subgroup of order four, hence is characteristic in . Further, is a complement to in .
- is complemented normal in : Since has index two, it is normal in (subgroup of index two is normal, or alternatively, nilpotent implies every maximal subgroup is normal). Further, the two-element subgroup is a permutable complement to in .
- is not complemented normal in : If were complemented normal in , it would also (by fact (1)) be complemented normal in the intermediate subgroup , which is a cyclic group of order eight. But we know that is not complemented in , since any element of not in generates .