The quaternion group, denoted $Q_8$, has exactly 6 subgroups.
Understanding the Quaternion Group ($Q_8$)
The quaternion group $Q_8$ is a non-abelian group of order eight. It consists of eight elements: ${1, -1, i, -i, j, -j, k, -k}$. Its multiplication rules are defined by:
- $i^2 = j^2 = k^2 = ijk = -1$
- $-1$ commutes with all elements and $(-1)^2 = 1$.
From these fundamental relations, other products can be derived, such as:
- $ij = k$
- $ji = -k$
- $jk = i$
- $kj = -i$
- $ki = j$
- $ik = -j$
Subgroup Structure of $Q_8$
The quaternion group exhibits a unique and well-defined subgroup structure. A significant property of $Q_8$ is that every one of its abelian subgroups is cyclic. This means that any subgroup that satisfies the commutative property (abelian) can be generated by a single element. Consequently, the "abelian subgroups of maximum rank"—which correspond to the largest possible cyclic abelian subgroups within $Q_8$—are specifically the center of the group and the three distinct subgroups of order four. The union of these key subgroups collectively generates the entire quaternion group.
Let's enumerate each of these subgroups based on their order:
-
The Trivial Subgroup
- Order: 1
- Description: This subgroup contains only the identity element.
- Subgroup: ${1}$
- Number of Subgroups: 1
-
The Center of the Group
- Order: 2
- Description: The center of $Q_8$, denoted $Z(Q_8)$, consists of elements that commute with every other element in the group. This subgroup is cyclic and is normal in $Q_8$.
- Subgroup: ${1, -1}$ (generated by $-1$, i.e., $\langle -1 \rangle$)
- Number of Subgroups: 1
-
Cyclic Subgroups of Order 4
- Order: 4
- Description: There are three distinct cyclic subgroups of order four. Each of these subgroups contains the center ${1, -1}$ and is generated by one of the "imaginary" units ($i$, $j$, or $k$). All three are normal subgroups of $Q_8$. As mentioned, these are considered abelian subgroups of maximum rank alongside the center.
- Subgroups:
- $\langle i \rangle = {1, -1, i, -i}$
- $\langle j \rangle = {1, -1, j, -j}$
- $\langle k \rangle = {1, -1, k, -k}$
- Number of Subgroups: 3
-
The Group Itself
- Order: 8
- Description: The quaternion group is, by definition, a subgroup of itself.
- Subgroup: $Q_8$
- Number of Subgroups: 1
Summary of Subgroups in $Q_8$
The following table provides a clear overview of all the subgroups of the quaternion group:
| Order | Subgroup(s) | Generators | Count | Properties |
|---|---|---|---|---|
| 1 | ${1}$ | $1$ | 1 | Trivial subgroup. |
| 2 | ${1, -1}$ | $-1$ | 1 | The unique subgroup of order 2. This is the center ($Z(Q_8)$), is cyclic, normal, and considered an abelian subgroup of maximum rank. |
| 4 | ${1, -1, i, -i}$ | $i$ | 3 | These three subgroups are distinct. They are all cyclic, abelian, and normal in $Q_8$. They are also considered abelian subgroups of maximum rank, and their join (along with the center) forms the whole group. |
| ${1, -1, j, -j}$ | $j$ | |||
| ${1, -1, k, -k}$ | $k$ | |||
| 8 | $Q_8$ | $i, j$ | 1 | The group itself. |
By summing the counts from each order, we find that the quaternion group $Q_8$ has a total of $1 + 1 + 3 + 1 = 6$ subgroups.
