Cyclic subgroup is characteristic in dihedral group

From Groupprops
Jump to: navigation, search
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 G be a dihedral group defined as follows:

G = \langle a,x \mid a^n = x^2 = e, xax^{-1} = a^{-1} \rangle

or the infinite dihedral group:

G = \langle a,x \mid x^2 = e, xax^{-1} = a^{-1} \rangle.

Then, for n \ge 3 or for the infinite dihedral group, the cyclic subgroup \langle a \rangle is a characteristic subgroup of G.

Related facts

Stronger facts

Proof

Let H = \langle a \rangle. Then, H has index two in G. Any element of G outside H is of the form a^kx. Further, (a^kx)^2 = a^kxa^kx^{-1} = a^ka^{-k} = e. Thus, every element of G outside of H has order two.

Thus, if n \ge 3, H can be defined as the unique cyclic subgroup generated by an element of order n.