Loading [MathJax]/jax/output/CommonHTML/jax.js

Abelian groups

Abbreviation: AbGrp nbsp nbsp nbsp nbsp nbsp Abelian group

Definition

An \emph{abelian group} is a structure G=G,+,,0, where + is an infix binary operation, called the \emph{group addition}, is a prefix unary operation, called the \emph{group negative} and 0 is a constant (nullary operation), called the \emph{additive identity element}, such that

+ is commutative: x+y=y+x

+ is associative: (x+y)+z=x+(y+z)

0 is an additive identity for +: 0+x=x

gives an additive inverse for +: x+x=0

Morphisms

Let G and H be abelian groups. A morphism from G to H is a function h:GH that is a homomorphism: h(x+y)=h(x)+h(y)

Remark: It follows that h(x)=h(x) and h(0)=0.

Examples

Example 1: Z,+,,0, the integers, with addition, unary subtraction, and zero. The variety of abelian groups is generated by this algebra.

Example 2: Zn=Z/nZ,+n,n,0+nZ, integers mod n.

Example 3: Any one-generated subgroup of a group.

Basic results

The free abelian group on n generators is Zn.

Classification of finitely generated abelian groups: Every n-generated abelian group is isomorphic to a direct product of Zpkii for i=1,,m and nm copies of Z, where the pi are (not necessarily distinct) primes and m0.

Properties

Classtype variety
Equational theory decidable in polynomial time
Quasiequational theory decidable
First-order theory decidable 1)
Locally finite no
Residual size ω
Congruence distributive no (Z2×Z2)
Congruence n-permutable yes, n=2, p(x,y,z)=xy+z
Congruence regular yes, congruences are determined by subalgebras
Congruence uniform yes
Congruence types permutational
Congruence extension property yes, if KHG then KG
Definable principal congruences no
Equationally def. pr. cong. no
Amalgamation property yes
Strong amalgamation property yes

Finite members

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
# of algs 1 1 1 2 1 1 1 3 2 1 1 2 1 1 1 5 1 2 1 1 1 1 1 3 2
# of si's 0 1 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 1 0 0 0 1 0 1

see also http://www.research.att.com/projects/OEIS?Anum=A000688

Subclasses

Superclasses

References


1) W. Szmielew, \emph{Decision problem in group theory}, Library of the Tenth International Congress of Philosophy, Amsterdam, August 11–18, 1948, Vol.1, Proceedings of the Congress, 1949, 763–766 MRreview

QR Code
QR Code abelian_groups (generated for current page)