Groups of Order 8
Theorem
Let $G$ be a group of order $8$.
Then $G$ is isomorphic to one of the following:
- $\Z_8$
- $\Z_4 \oplus \Z_2$
- $\Z_2 \oplus \Z_2 \oplus \Z_2$
- $D_4$
- $\Dic 2$
where:
- $\Z_n$ is the cyclic group of order $n$
- $D_4$ is the dihedral group of order $8$
- $\Dic 2$ is the dicyclic group of order $8$.
Proof
The abelian cases are handled by the corollary to Abelian Group Factored by Prime.
$\Box$
Let $G$ be non-abelian.
By Lagrange's theorem the order of non-identity elements in $G$ is either $2$, $4$ or $8$.
Aiming for a contradiction, suppose that there exists an order $8$ element.
Then $G$ is generated by this element.
So $G$ is by definition cyclic.
But Cyclic Group is Abelian, contradicting the assumption that $G$ is non-abelian.
So there is no order $8$ element.
By Non-Abelian Order 8 Group has Order 4 Element, there exists at least one order $4$ element in $G$.
Let it be denoted by $a$.
Let $A$ denote the subgroup generated by $a$.
By Lagrange's theorem there are two cosets in $G$: $A$ and $G \setminus A$.
Let $b \in G \setminus A$.
Then $\set {a, b}$ is a generator of $G$.
Now we consider how $a$ and $b$ interact with each other.
Consider the element $x = b a b^{-1}$.
By Subgroup of Index 2 is Normal, $b A b^{-1} = A$.
So $x \in A$.
By Order of Conjugate Element equals Order of Element, the only possible choices are $x = a$ or $x = a^3$.
If $x = a$, then $a$ and $b$ commute.
Since $\set {a, b}$ generates $G$, this makes $G$ an abelian group, which is a contradiction.
So $b a b^{-1} = a^3$.
It suffices to consider the order of $b$:
- If $\order b = 2$, then $G \cong D_4$.
- If $\order b = 4$, then $G \cong \Dic 2$.
$\blacksquare$
Also see
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 62 \gamma$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Groups with eight elements