In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that

a * b = b * a

for all a and b in G. In other words, the order of elements in a product doesn’t matter. Such groups are generally easier to understand.

Abelian groups are named after Niels Henrik Abel. Groups that are not commutative are called non-abelian (rather than non-commutative).

From Wikipedia

For Ron and Douglas, who surely understand this joke:

Abelian groups

What’s abelian and purple? An abelian grape!

Definition 5.6.1 Let and be two elements of a group . We say that commutes with (or that commute) if . We call a group commutative (or abelian) if every pair of elements belonging to commute. If is a group which is not necessarily commutative then we call noncommutative (or nonabelian).

Now the reader should understand the punchline to the joke quoted at the beginning! (Let uncontrolled laughter ensue!)

From some smart Navy site

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