Center of dihedral group

The center of the dihedral group $D_{2n}=\langle x,a\mid a^{n}=x^{2}=e,xax^{-1}=a^{-1}\rangle$ (of order $2n$), denoted $Z(D_{2n})$, is the subgroup of $D_{2n}$ made from the set of elements in the group that commute with all other elements.

Result

When $n$ is even, $Z(D_{2n})=\{e,a^{n/2}\}$, isomorphic to the cyclic group:Z2.

When $n$ is odd, $Z(D_{2n})=\{e\}$, isomorphic to the trivial group.

Pages on particular cases

• Center of dihedral group:D8
• Center of dihedral group:D16