Groups of Order 30/Lemma

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

Then $G$ is one of the following:


 * The cyclic group $C_{30}$


 * The dihedral group $D_{15}$


 * Isomorphic to one of:


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


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

Proof
By Group of Order 30 has Normal Cyclic Subgroup of Order 15, $G$ has a normal subgroup of order $15$ which is cyclic.

Let this normal cyclic order $15$ subgroup be denoted $N$:


 * $N = \gen x$

Let $y$ be the generator for any Sylow $2$-subgroup of $G$.

Then:

Then:

and so:
 * $i^2 - 1 \equiv 0 \pmod {15}$

Investigating the powers of $i$, case by case, searching for those which satisfy this congruence, yields:
 * $i \in \set {1, 4, 11, 14}$

The case $i \equiv 1 \pmod {15}$ leads to the cyclic group $C_{30}$.

The case where $i \equiv {14} \equiv {-1} \pmod {15}$ leads to the dihedral group $D_{15}$.

The other two cases lead to:
 * $\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$


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