Table of Contents
Abelian groups
Abbreviation: AbGrp nbsp nbsp nbsp nbsp nbsp 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 1) |
| 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
