Axiomatic WikiWWikiDDefinitionsAbelian group
Definition·D120
A group whose operation is commutative: the order of the operands does not matter.
An abelian group is a group
such that
In words
Let
be an abelian group, then for any a, b, if a and b lie in G then a star b equals b star a (i.e. commutative).
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 · A09 (computed from the citation graph, not asserted).
Remarks
Also called a commutative group. Commutativity is a fourth axiom on top of the three group axioms (G1)-(G3) of D038: it is independent, not implied by associativity, identity, and inverses. A group whose operation is not commutative is called non-abelian; the symmetric group on three or more letters is the standard example.Two notational conventions are common. Additive notation writes the operation as
, the identity as
, the inverse of
as
, and repeated application as
; it is the default for modules and rings, and is often used to emphasize that a group is abelian. Multiplicative notation writes the operation as
or juxtaposition, the identity as
or
, the inverse as
, and repeated application as
; it is the default for groups in general.The archetypal examples are
(T22) and
(T26); a finite example is the multiplicative units modulo
(T61). In an abelian group every subgroup is normal (D048), so quotients are unrestricted.