Dihedral Group D4/Group Presentation

Group Presentation of Dihedral Group $D_4$

The group presentation of the dihedral group $D_4$ is given by:

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

Proof

We have that the group presentation of the dihedral group $D_n$ is:

$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$

Setting $n = 4, \alpha = a, \beta = b$, we get:

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

from which the result follows.

$\blacksquare$