Quaternion group

Description

The quaternion group $Q_8$ is the set $\{1, -1, i, -i, j, -j, k, -k\}$ of quaternions, with quaternion multiplication as the operation. The multiplication follows from the rules $i^2 = j^2 = k^2 = ijk = -1$, where $-1$ commutes with each element. These rules give $ij = k$, $jk = i$, $ki = j$ and $ji = -k$, $kj = -i$, $ik = -j$. The group has order $8$, and it is not abelian.

Remarks

The element $-1$ is the only element of order $2$, and the six elements $\pm i, \pm j, \pm k$ have order $4$. The centre and the commutator subgroup of $Q_8$ are both $\{1, -1\}$, and $Q_8 / \{1, -1\}$ is the Klein four-group. Each subgroup of $Q_8$ is normal, although $Q_8$ is not abelian. The groups $Q_8$ and $D_4$ are not isomorphic, because $D_4$ has five elements of order $2$.

Explanations

L1 human

The quaternion group is a small group of order $8$ in which the elements do not commute. You can think of $i$, $j$, $k$ as three square roots of $-1$ whose products go around in a circle. It is a good test case, because many simple guesses about groups fail for it.