# Definition:Order of Structure

## 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

Other notations that can sometimes be seen for the order of an algebraic structure $S$ include:

$\map \# S$
$\operatorname {order} S$

Some sources use $\map o S$, 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.