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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:G→H 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 n−m copies of Z, where the pi are (not necessarily distinct) primes and m≥0.
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)=x−y+z |
Congruence regular | yes, congruences are determined by subalgebras |
Congruence uniform | yes |
Congruence types | permutational |
Congruence extension property | yes, if K≤H≤G then K≤G |
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