In mathematics, particularly in the field of abstract algebra, a finite group is a group that consists of a finite number of elements under a binary operation satisfying closure, associativity, the existence of an identity element, and inverses for each element. The number of elements in such a group$ G $,$G$,denoted$ |G| $,$|G|$,is called the order of the group.

A group is defined as a nonempty set$ G $$G$together with a binary operation$ *$*: G \times G \toG $G$satisfying four fundamental axioms: closure, associativity, the existence of an identity element, and the existence of inverses. This definition, introduced in the 19th century and refined over time, captures the essence of reversible operations while excluding structures that fail to maintain these properties.

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