Groups of order $2p$ are the most concrete family among groups of order $pq$. When $p$ is an odd prime, the group is either cyclic or has the symmetry pattern of a regular $p$-gon. In this lesson, we focus in the dihedral groups and groups of order 2p. The discussion links Sylow theory with the familiar presentation of a dihedral group. Students often remember the relation $ba=a^{-1}b$ but forget that it comes from a conjugation action on a normal cyclic subgroup.

Let $n\geq 3$ be an integer. A group $D_n$ is called a \textbf{dihedral group of degree} $n$ if it is generated by elements $a$ and $b$ such that $$\begin{equation*} a^n=e,\qquad b^2=e,\qquad ba=a^{-1}b. \end{equation*}$$