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Mathematics-Online lexicon:

Group


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

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 $ G$ is written instead of $ (G,\diamond)$.
(Authors: Burkhardt/Höllig/Hörner)

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  automatically generated 3/31/2005