Cosets and finite groups of orthogonal operators on the plane

Question

Can someone show me how to compute the left cosets of the subgroup $H=\{1, x^5\}$ in the Dihedral group $D_{10}$

I know that $D_{10}$ is generated by two elements $x$ and $y$ such that: $$x^{10}=1, y^2=1 \text{ and } yx=x^{-1}y$$

Answer 1

This is -essentially- a straight forward calculation, using the fact that for two elements $a, b\in D_{10}$ we have that $a\in bH$ if and only if $b^{−1}a\in H$. Each coset contains precisely two elements (why?)

For example, $xH=x^6H$, because $x^6\in xH$ as $x^{-1}x^6=x^5\in H$. Then because each coset contains two elements we have that $xH=\{x, x^6\}$.