# Definition:Order of Structure

## Contents

## Definition

The **order** of an algebraic structure $\struct {S, \circ}$ is the cardinality of its underlying set, and is denoted $\order S$.

Thus, for a finite set $S$, the **order of $\struct {S, \circ}$** is the number of elements in $S$.

### Infinite Structure

Let the underlying set $S$ of $\struct {S, \circ}$ be infinite.

Then $\struct {S, \circ}$ is an **infinite structure**.

### Finite Structure

Let the underlying set $S$ of $\struct {S, \circ}$ be finite.

Then $\struct {S, \circ}$ a **finite structure**.

## Also defined as

Some sources do not define the **order of a structure** for an underlying set of infinite cardinality, restricting themselves to the finite case.

## Notation

Some sources use $\map o S$ for the order of $S$, but this has problems of ambiguity with other uses of $\map o n$. (See little-o notation.)

## Also see

This definition is mostly used in the context of group theory:

- Results about
**the order of a group**can be found here.

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): $\S 4$: Alternative Axioms for Finite Groups - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.4$. Gruppoids, semigroups and groups - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.4$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 25$ - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.4$ - 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): Introduction: Notation - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{II}$: Problem $\text{GG}$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 38$ - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 34$. Examples of groups: $(6)$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Proposition $5.8$ Notation

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 5$: Groups $\text{I}$ - 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups