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  "body": "\n## Description\n\nThe quaternion group $Q_8$ is the set $\\{1, -1, i, -i, j, -j, k, -k\\}$ of quaternions, with quaternion multiplication as the operation.\nThe multiplication follows from the rules $i^2 = j^2 = k^2 = ijk = -1$, where $-1$ commutes with each element.\nThese rules give $ij = k$, $jk = i$, $ki = j$ and $ji = -k$, $kj = -i$, $ik = -j$.\nThe group has order $8$, and it is not [[abelian|082GHNPJ53]].\n\n## Remarks\n\nThe element $-1$ is the only element of order $2$, and the six elements $\\pm i, \\pm j, \\pm k$ have order $4$.\nThe [[centre|0DZ93493VN]] and the [[commutator subgroup|0ASGSTXSBW]] of $Q_8$ are both $\\{1, -1\\}$, and $Q_8 / \\{1, -1\\}$ is the [[Klein four-group|0PMRQJVTME]].\nEach subgroup of $Q_8$ is [[normal|0CB0DX5BVV]], although $Q_8$ is not abelian.\nThe groups $Q_8$ and $D_4$ are not isomorphic, because $D_4$ has five elements of order $2$.\n\n## Explanations\n\n### L1 human\n\nThe quaternion group is a small group of order $8$ in which the elements do not commute.\nYou can think of $i$, $j$, $k$ as three square roots of $-1$ whose products go around in a circle.\nIt is a good test case, because many simple guesses about groups fail for it.\n",
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