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Abelian (Commutative) Groups

A group with the commutative property:

$\displaystyle a\circ b = b\circ a $

is called either a commutative group (which is obvious) or an abelian group (which is not so obvious, but you should know what this word means). Note well! Not all groups are abelian! In particular, the rotation group SO(3) (discussed below) is nonabelian, because two rotations through a finite angle around two distinct axes do not produces the same final coordinate frame when performed in either order. Many if not most of the transformation groups of physics are non-abelian, and they play an extremely important role in quantum theory.

Robert G. Brown 2017-07-11