Abelian group

Group

A group $(G,\oplus )$ is a set G together with an operation

$\oplus :G\times G\longrightarrow G$

with the following properties:

Identity

There is a unique element 0 in G such that for all elements g in G, $g\oplus \mathbf {0} =g$ and $\mathbf {0} \oplus g=g$.

Inverse

For every element g in G there is a unique element ${}^{\ominus }\!g$ such that $g\oplus {}^{\ominus }\!g=\mathbf {0}$ and ${}^{\ominus }\!g\oplus g=\mathbf {0}$.

Associativity

For any three elements f, g, and h in G,

$f\oplus (g\oplus h)=(f\oplus g)\oplus h$.

Abelian group

Niels Henrik Abel
(5 Aug 1802 – 6 Apr 1829)

A group $(G,\oplus )$ having the foregoing properties of identity, inverse and associativity is further said to be Abelian if its group operation $\oplus$ is commutative.

Commutativity

For any two elements g and h in G,

$h\oplus g=g\oplus h$.