Groups Up To Order Eight

We classify all groups with at most eight elements. Recall groups of prime order are cyclic, so we need only focus on the cases $|G| = 4,6,8$. We make use of the following:

Lemma: If each element $1 \ne g\in G$ is of order 2, then $G$ is abelian and isomorphic to $\mathbb{Z}_2 \times ... \times \mathbb{Z}_2$ and $|G|$ is a power of 2.

Proof: Clearly true for $|G| = 2$. Otherwise, let $1 \ne a \ne b \in G$. We have $a^2 = b^2 = 1$, that is $a = a^{-1}, b=b^{-1}$. Then $a b \ne 1$ (otherwise $a = b^{-1} = b$) and $1 = (a b)^2 = a (b a)b$ which implies $b a = a^{-1} b^{-1} = a b$. Thus $G$ is abelian.

Since $G$ is finite, it has a finite set of independent generators $a_1,...,a_n$. As $G$ abelian, we may write an element $g \in G$ in the form

\[ g = a_1^{e_1} ... a_n^{e_n} \]

where each $e_i \in\{0,1\}$. Then $G = \langle a_1 \rangle \times ... \times \langle a_n \rangle$ and $|G| = 2 \times ... \times 2 = 2^n$

Now we can classify the groups up to order eight:

  • $|G| = 4$: Each element (besides the identity) must have order 2 or 4. If $a \in G$ has order 4 it generates $G$ and we have $G = \mathbb{Z}_4$. Otherwise every element has order 2 and by the lemma we have $G = \mathbb{Z}_2 \times \mathbb{Z}_2$ (the four-group or quadratic group, sometimes denoted by $V$ after F. Klein’s "Vierergruppe").

  • $|G| = 6$: If $a \in G$ has order 6 we have $G = \mathbb{Z}_6$. Otherwise all elements (besides the identity) have order 2 or 3. By the lemma, not all elements can have order 2 because 6 is not a power of 2. So let $a$ be an element of order 3, that is $1,a,a^2$ are distinct. Let $b$ be some other element in $G$. It can be verified that $1,a,a^2,b,a b,a^2 b$ must be distinct. In order to satisfy closure, $b^2$ must be one of these elements. The only possibilities are $b^2 = 1,a$ or $a^2$.

    If $b^2=a,a^2$ we find that $b$ cannot have order 2, so it has order 3. Then $1 = a b$ or $1 =a^2 b$, both of which are contradictions. Hence $b^2 = 1$. Next we determine which element is equal to $b a$. The only possible choices are $a b$ or $a^2 b$. If $b a = a b$, then $G$ is abelian, but then $(a b)^2 = a^2$ and $(a b)^3 = b$ implying that $a b$ has order 6, a contradiction. Thus $b a = a^2 b$, implying $(a b)^2 = 1$. We have defining relations $a^3 = b^2 = (a b)^2 = 1$. We shall see later that this is indeed a group (associativity turns out to hold) because it is the symmetric group of degree 3 (which is isomorphic to the dihedral group of order 6).

  • $|G|=8$: It turns out there are 3 abelian groups and 2 nonabelian groups. The three abelian groups are easy to classify: $\mathbb{Z}_8, \mathbb{Z}_4 \times \mathbb{Z}_2, \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$.

    The other groups must have the maximum order of any element greater than 2 but less than 8. Hence there exists an element of order 4, which we denote by $a$. All the others (besides the identity) have order 2 or 4. Let $b$ be an element not generated by $a$. Then we have the distinct elements $1,a,a^2,a^3,b,a b,a^2 b,a^3 b$. Now $b^2$ can only be one of the first four. But $b^2 = a, a^3$ imply $b$ is not of order 2 or 4, so we must have $b^2 =1$ or $b^2= a^2$.

    Suppose $b^2 = 1$. Now $b a$ must be equal to one of the last three elements. If $b a = a b$ then the group is abelian and we end up with the aforementioned $\mathbb{Z}_4 \times \mathbb{Z}_2$. If $b a = a^2 b$, then we have $b^{-1}a^2 b = a$. Upon squaring, we derive the contradictory $a^2 = 1$. So we must have $b a = a^3 b$, that is, $(a b)^2 = 1$. The defining relations are $a^4 = b^2 = (a b)^2 = 1$, and this turns out to be the dihedral group of order 8, also known as the octic group.

    The other possibility is $b^2 = a^2$. In this case, $b$ also has order 4. If $b a = a b$ then the group is abelian and again we wind up with the group $\mathbb{Z}_4 \times \mathbb{Z}_2$. If $b a = a^2 b$ we have $b a = b^3$, which is a contradiction because it implies $a = b^2 = a^2$. Thus we must have $b a = a^3 b$. Then we get a group with the defining relations $a^4 = 1, a^2 = b^2, ba = a^3 b$, which is known as the quaternion group. To verify associativity, one can show it is isomorphic to the group generated by the matrices

    \[ \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]

    or

    \[ \begin{pmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix} \]

    The quaternion group is a special case of a dicyclic group, groups of order $4 m$ given by $a^{2m} = 1, a^m = (a b)^2 = b^2$, and whose elements can be written $1,a,...,a^{2m-1},b,a b,...,a^{2m -1}b$. The square of elements not generated by $a$ is $b^2$.