structural:expansion, quality:correction
Definition

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.

Richard Borcherds, Mathematicians: An Outer View of the Inner World

A group is a non-empty set G {\displaystyle G} together with a binary operation on G {\displaystyle G}, here denoted " ⋅ {\displaystyle \cdot } ", that combines any two elements a {\displaystyle a} and b {\displaystyle b} of G {\displaystyle G} to form an element of G {\displaystyle G}, denoted a ⋅ b {\displaystyle a\cdot b}, such that the following three requirements, known as group axioms, are satisfied: