Groups of Order 30

Theorem
Let $G$ be a group of order $30$.

Then $G$ is one of the following:


 * The cyclic group $C_{30}$


 * The dihedral group $D_{15}$


 * The group direct product $C_5 \times D_3$


 * The group direct product $C_3 \times D_5$

Proof
First we introduce a lemma:

Lemma
From the lemma, it remains to be shown that the group presentations:


 * $\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^{11} }$


 * $\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$

give the groups $C_5 \times D_3$ and $C_3 \times D_5$.