=====Abelian groups=====

Abbreviation: **AbGrp** nbsp nbsp nbsp nbsp nbsp [[wp>Abelian group]]

====Definition====
An \emph{abelian group} is a structure $\mathbf{G}=\langle
G,+,-,0\rangle$, 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 $\mathbf{G}$ and $\mathbf{H}$ be abelian groups. A morphism from $\mathbf{G}$ to $\mathbf{H}$ is a function $h:G\rightarrow H$ 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: $\langle \mathbb{Z}, +, -, 0\rangle$, the integers, with addition, unary subtraction, and zero. The variety of abelian groups is generated by this algebra.

Example 2: $\mathbb Z_n=\langle \mathbb{Z}/n\mathbb Z, +_n, -_n, 0+n\mathbb Z\rangle$, integers mod $n$.

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


===Basic results===
The free abelian group on $n$ generators is $\mathbb Z^n$.

Classification of finitely generated abelian groups: Every $n$-generated abelian group is isomorphic to a direct product of $\mathbb Z_{p_i^{k_i}}$ for $i=1,\ldots,m$ and $n-m$ copies of $\mathbb Z$, where the $p_i$ are (not necessarily distinct) primes and $m\ge 0$.


====Properties====
^[[Classtype]]                       |variety |
^[[Equational theory]]               |decidable in polynomial time |
^[[Quasiequational theory]]          |decidable |
^[[First-order theory]]              |decidable [(Szmielew1949)] |
^[[Locally finite]]                  |no |
^[[Residual size]]                   |$\omega$ |
^[[Congruence distributive]]         |no ($\mathbb{Z}_{2}\times \mathbb{Z}_{2}$) |
^[[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 $K\le H\le G$ then $K\le G$ |
^[[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


====Subclasses====
[[Boolean groups]]

[[Commutative rings]]


====Superclasses====
[[Groups]]

[[Commutative monoids]]


====References====

[(Szmielew1949>
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 [[http://www.ams.org/mathscinet-getitem?mr=10:500a|MRreview]])]