Definition:Order of Structure
This page is about order of structure. For other uses, see order.
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.
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 4$: Alternative Axioms for Finite Groups
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.4$. Gruppoids, semigroups and groups
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem
- 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): Chapter $\text{II}$: Groups: A Little Number Theory
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Problem $\text{GG}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 38$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 5$: Groups $\text{I}$
- 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $2$. GROUP: Example $2$
- 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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Lagrange's theorem: 2.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): order: 6. (of a group or element)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Lagrange's theorem: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order: 6. (of a group or element)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): order (of a group)
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- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups