📂Abstract Algebra

Dihedral Group in Abstract Algebra

Definition ¹

The subgroup $D_{n} \leqslant S_{n}$ of the symmetric group is defined as the group consisting only of the permutations that rotate and reflect an $n$-gon, and it is called the dihedral group.

Explanation

Since it is derived from figures, it is hard to explain with words alone.

$D_{3} = S_{3}$

As the smallest example of a dihedral group, there is the symmetric group $D_{3} = S_{3}$.

$| D_{n} | =2n$

It is not hard to guess that there exist $2n$ such permutations for an $n$-gon. For instance, $D_{4}$, built on a square, has $8$ elements and is hence also known by the nickname octic group.

As shown in the figure above, the elements of $D_{4}$ are $\mu_{1}, \mu_{2}, \delta_{1}, \delta_{2}$ and the rotations $\rho_{0} , \rho_{1} , \rho_{2} , \rho_{3}$.