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## 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

**, $-$ is a prefix unary operation, called the**

*group addition***and $0$ is a constant (nullary operation), called the**

*group negative***, such that**

*additive identity element*$+$ 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,

**, Library of the Tenth International Congress of Philosophy, Amsterdam, August 11–18, 1948, Vol.**

*Decision problem in group theory***1**, Proceedings of the Congress, 1949, 763–766 MRreview

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