Definition:Order of a 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.)

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.4$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 25$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.4$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 38$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 34 \ (6)$
• John F. Humphreys: A Course in Group Theory (1996): $\S 5$: Proposition $5.8$: Notation