Abelian groups

Abbreviation: AbGrp                     Abelian group

Definition

An abelian group is a structure $\mathbf{G}=\langle G,+,-,0\rangle$, where $+$ is an infix binary operation, called the group addition, $-$ is a prefix unary operation, called the group negative and $0$ is a constant (nullary operation), called the 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

Subclasses

Superclasses

References


1) W. Szmielew, 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