Isomorphism of Subgroup of $D_n$ If $k|n$ , $k \ge 2$ I am trying to show that $D_n$ has a subgroup isomorphic to $D_k$. I know that by Lagrange's theorem If a subgroup of order $k$  exists in $D_n$ it will divide $2n$ then $k$ divides both $n$ and $2n$, but how can I show such a subgroup exists and that it is isomorphic to $D_k$, i.e that it is generated by $2$ elements say $a$ and $b$ where $o(a) = k$, $o(b) = 2$ and $aba = b$. Thanks any help much appreciated.

Use the geometric definition of $D_n$ as the symmetry group of the regular $n$-gon. It is then clear that if you connect every $n/k$ vertex of  the vertices of the regular $n$-gon you get a regular $k$-gon. Every rotation or reflection that leaves the regular $k$-gon invariant also leaves $n$-gon invariant, because they are rigid transformations.
