Group (mathematics)

The axioms for a group are short and natural... Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.

Given this set of symmetries and the described operation, the group axioms can be understood as follows.

The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers.

The desire for the existence of multiplicative inverses suggests considering fractions

Any finite abelian group is isomorphic to a product of finite cyclic groups; this statement is part of the .