# Cyclic subgroup is characteristic in dihedral group

From Groupprops

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup satisfying a particular subgroup property (namely, Characteristic subgroup (?)) in a particular group or type of group (namely, Dihedral group (?)).

## Statement

Let be a dihedral group defined as follows:

or the infinite dihedral group:

.

Then, for or for the infinite dihedral group, the cyclic subgroup is a characteristic subgroup of .

## Related facts

### Stronger facts

- Cyclic subgroup is isomorph-free in dihedral group
- Cyclic subgroup is prehomomorph-contained in dihedral group

## Proof

Let . Then, has index two in . Any element of outside is of the form . Further, . Thus, every element of outside of has order two.

Thus, if , can be defined as the unique cyclic subgroup generated by an element of order .