Mathematics-Online lexicon: Group

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	Mathematics-Online lexicon:
	Group

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A group $(G,\diamond)$ is a set together with a binary operation $\diamond$ :

$\displaystyle \diamond: G \times G \longmapsto G\,,$

that is, a uniquely determined element $a \diamond b \in G$ is assigned to each pair of elements

where $a,b \in G$ . Furthermore the operation must satisfy the following requirements (group axioms):

associativity:

$\displaystyle \forall a,b,c \in G \quad (a \diamond b) \diamond c = a \diamond (b \diamond c)$
There exists a neutral element (or identity element):

$\displaystyle \exists e \in G \quad \forall a \in G \quad e \diamond a = a \diamond e = a$
It can be proved that the identity element is uniquely determined.
There exists a reciprocal for each element of :

$\displaystyle \forall a \in G \quad \exists b \in G \quad a \diamond b = b \diamond a = e$
The element is called inverse element of . This inverse element is uniquely determined and is often denoted by $a^{-1}$ .

A group is called commutative or Abelian group, if its operation is commutative:

$\displaystyle \forall a,b \in G \quad a \diamond b = b \diamond a$

If it is clear which operation is used, then often only

is written instead of $(G,\diamond)$ .

(Authors: Burkhardt/Höllig/Hörner)

automatically generated 3/31/2005