Abelian groups arising from monoidal operations

This article discusses a general pattern that gives rise to Abelian groups in diverse branches of mathematics, including, for instance, the Brauer group of a field, the ideal class group, and the group structure on vector bundles over a manifold.

There are the following general steps:


 * We identify a binary operation, like a sum or product. Sometimes, this operation is commutative and associative on the nose, while at other times, it is commutative and associative up to natural isomorphism or equivalence.
 * We identify a neutral element (identity element) for this binary operation. If such an element does not exist, we add it in. The upshot: we have an Abelian monoid.
 * Next, we identify certain elements that can be called principal or trivial. The method for identifying principal or trivial elements depends on the context.
 * Finally, we verify that for any element in the monoid, there exists an element such that their sum or product is trivial. Thus, quotienting out by the trivial elements yields an Abelian group.

A key point to remember in this procedure is that the binary operation as initially defined may appear very far from a group -- it may appear that adding or multiplying two elements makes them irreversibly larger or smaller, thus making inverses impossible. In all these cases, it is an appropriate identification of trivial (or principal) elements that allows quotienting out to obtain a group.