G'day all.
Quoting Rik van Ginneken
It is more even subtle if one considers the rotation group. The unit is keeping an object on its place. The multiplication is doing rotations sequently. Allright, one has an inverse here, but the rule (a*b)*c = a*(b*c) doesn't occur; try it with a banana or an apple!
Wrong. Rotations do indeed form a group. In three dimensions, it's isomorphic to the unit quaternion group. I think you're thinking of unit octonions, which don't form a group because octonion multiplication is not associative in general.
Please make a distinction betwixt "monoids" and "monads". The first is a group without an invertion.
...or a semigroup with an identity.
The latter is a construction in category theory (=abstract nonsense),
Them's fighting words! Cheers, Andrew Bromage