Dihedral Group D4/Center

Center of the Dihedral Group $D_4$

Let $D_4$ denote the dihedral group $D_4$, whose group presentation is given as:

$D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$

The center of $D_4$ is given by:

$\map Z {D_4} = \set {e, a^2}$

Proof

From Center of Dihedral Group:

$\map Z {D_n} = \begin{cases} e & : n \text { odd} \\ \set {e, \alpha^{n / 2} } & : n \text { even} \end{cases}$

Hence the result.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Exercise $5$