THEOREM OF THE DAY: Moufang’s Theorem

In a Moufang loop any three elements which associate generate a group.

With i the imaginary constant whose square is $−1$, the set ${±1, ±i}$ forms a group: multiplication keeps you inside the set, it allows inverses (e.g. $i × −i = −i^ 2 = −(−1) = 1$, so $i^{ −1} = −i$) and it is associative (that is, $x × (y × z)$ is the same as $( x × y) × z$ — the bracketing can safely be forgotten). In $1843$, the great Irish scientist William Rowan Hamilton discovered the quaternions: i is joined by mysterious companions j and k who multiply according to the circular diagram above left: if $x$ and $y$ follow each other clockwise round the circle, then x × y = + the other quantity; if anticlockwise, the product is negative: $i j = k$, $k j = −i$, etc.

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