(Solved): 4. Label the vertices of a regular hexagon from 1 to 6 in clockwise order. Consider the three major ...

4. Label the vertices of a regular hexagon from 1 to 6 in clockwise order. Consider the three major diagonals: $L_{14}$ going through the vertices 1 and 4 , $L_{25}$ going through the vertices 2 and 5 , and $L_{36}$ going through the vertices 3 and 6 . (a) Find the set $H$ of elements of $D_6$ which fix the diagonal $L_{14}$. (Warning: to fix the diagonal $L_{14}$, the element of $D_6$ can either fix the vertices 1 and 4 , or swap the vertices 1 and 4.) Prove that this subset $H$ is a subgroup. (b) Find a set of coset representatives for $H$ in $D_6$. Can you give a "natural" bijection between the following sets: $\left\{L_{14},L_{25},L_{36}\right\}\to \{\text{ cosets of }H\}?$ That is, a bijection between the set of major diagonals and the set of cosets of $H$. Here, we would like you to think about what "natural" might mean.