Definition:Order of Structure

Definition
The order of an algebraic structure $$\left({S, \circ}\right)$$ is the cardinality of its underlying set, and is denoted $$\left|{S}\right|$$.

That is, the order of $$\left({S, \circ}\right)$$ is the number of elements in $$S$$.

If the set $$S$$ is infinite, then $$\left({S, \circ}\right)$$ is of infinite order, or is described as an infinite structure.

Otherwise it is of finite order, or is described as a finite structure.

This definition is mostly used in the study of group theory.

Notation
Some sources use $$o \left({S}\right)$$ for the order of $$S$$, but this has problems of ambiguity with other uses of $$o \left({n}\right)$$. (See O-notation.)