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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\oplus\mathbf 0=g} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf 0\oplus g=g} .

Inverse

For every element g in G there is a unique element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}^\ominus\! g} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\oplus{}^\ominus\! g=\mathbf 0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}^\ominus\! g\oplus g=\mathbf 0} .

Associativity

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\oplus (g\oplus h) = (f\oplus g)\oplus h} .

Abelian group

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

A group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G,\oplus)} having the foregoing properties of identity, inverse and associativity is further said to be Abelian if its group operation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oplus} is commutative.

Communutativity

For any two elements g and h in G,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h\oplus g = g\oplus h} .