📂Abstract Algebra

Infinite Cyclic groups in Abstract Algebra

Definition ¹

A subgroup $D_{n} \leqslant S_{n}$ of the symmetric group comprised solely of permutations that rotate and reflect a $n$-sided polygon is defined as the Dihedral group.

Description

Since it is derived from geometrical figures, it’s hard to explain just with words.

$D_{3} = S_{3}$

An example of the smallest dihedral group is the symmetric group $D_{3} = S_{3}$.

$| D_{n} | =2n$

Such permutations for a $n$-sided polygon can relatively easily be guessed to exist in $2n$. For instance, the group based on a square $D_{4}$ has $8$ elements and is hence also known by the nickname Octic group.

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