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Abelian groups

Abbreviation: AbGrp 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

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

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