Definition:Abelian Group/Definition 1

Definition

An abelian group is a group $G$ where:

$\forall a, b \in G: a b = b a$

That is, every element in $G$ commutes with every other element in $G$.

Also known as

The usual way of spelling abelian group is without a capital letter, but Abelian is frequently seen.

The term commutative group can occasionally be seen.

Also see

• Equivalence of Definitions of Abelian Group

• Results about abelian groups can be found here.

Source of Name

This entry was named for Niels Henrik Abel.

Historical Note

The importance of abelian groups was discovered by Niels Henrik Abel during the course of his investigations into the theory of equations.

Linguistic Note

The pronunciation of abelian in the term abelian group is usually either a-bee-lee-an or a-bell-ee-an, putting the emphasis on the second syllable.

Note that the term abelian has thus phonetically lost the connection to its eponym Abel (correctly pronounced aah-bl).

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields
• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory: Definition $2$
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.4$. Gruppoids, semigroups and groups
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.3$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: The Group Property
• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring: Definitions $1.1 \ \text{(b)}$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 29$
• 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
• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups: Definition $1.1 \text{(iv)}$
• 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): $\S 33$. The definition of a group
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Abelian group
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): commutative or permutable: 2. (of a structure)
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Definition $1.3$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Abelian group
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): group
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Abelian group
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): group
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): abelian group
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): abelian group

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• 1992: William A. Adkins and Steven H. Weintraub: Algebra: An Approach via Module Theory ... (previous) ... (next): $\S 1.1$: Example $1$