Abbreviation: AbGrp nbsp nbsp nbsp nbsp nbsp Abelian group

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 $+$: $-x+x=0$

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 $h(0)=0$.

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.

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 $m\ge 0$.

see also http://www.research.att.com/projects/OEIS?Anum=A000688

Boolean groups

Commutative rings

Groups

Commutative monoids