## 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 decidable in polynomial time decidable decidable 1) no $\omega$ no ($\mathbb{Z}_{2}\times \mathbb{Z}_{2}$) yes, $n=2$, $p(x,y,z)=x-y+z$ yes, congruences are determined by subalgebras yes permutational yes, if $K\le H\le G$ then $K\le G$ no no yes yes

### Finite members

 $n$ # of algs # of si's 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 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 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

### 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