# Groups of Order 30/Lemma

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## 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:

\(\displaystyle y x y^{-1}\) | \(\in\) | \(\displaystyle N\) | as $N$ is normal | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle y x y^{-1}\) | \(=\) | \(\displaystyle x^i\) | for some $i \in \Z_{\ge 0}$ |

Then:

\(\displaystyle x\) | \(=\) | \(\displaystyle y^2 x y^{-2}\) | as $y^2 = e$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle y \paren {y x y^{-1} } y^{-1}\) | Group Axiom $\text G 1$: Associativity | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle y x^i y^{-1}\) | as $y x y^{-1} = x^i$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {y x y^{-1} }^i\) | Power of Conjugate equals Conjugate of Power | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {x^i}^i\) | as $y x y^{-1} = x^i$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x^{i^2}\) | Powers of Group Elements |

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:

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

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

$\blacksquare$

## Sources

- 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $12$: Applications of Sylow Theory: $(5)$ Groups of order $30$: Proposition $12.6$