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
.