Dihedral Group D4/Group Presentation
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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.
$\blacksquare$