Quaternion Group - Cayley Table

Cayley Table

The Cayley table (multiplication table) for Q is given by:

1 −1 i −i j −j k −k
1 1 −1 i −i j −j k −k
−1 −1 1 −i i −j j −k k
i i −i −1 1 k −k −j j
−i −i i 1 −1 −k k j −j
j j −j −k k −1 1 i −i
−j −j j k −k 1 −1 −i i
k k −k j −j −i i −1 1
−k −k k −j j i −i 1 −1

The multiplication of pairs of elements from the subset {±i, ±j, ±k} works like the cross product of unit vectors in three-dimensional Euclidean space.

\begin{alignat}{2} ij & = k, & ji & = -k, \\ jk & = i, & kj & = -i, \\ ki & = j, & ik & = -j. \end{alignat}

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