[home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff]

Mathematics-Online lexicon:

# Group

 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

A group is a set together with a binary operation :

that is, a uniquely determined element is assigned to each pair of elements where . Furthermore the operation must satisfy the following requirements (group axioms):
• associativity:

• There exists a neutral element (or identity element):

It can be proved that the identity element is uniquely determined.
• There exists a reciprocal for each element of :

The element is called inverse element of . This inverse element is uniquely determined and is often denoted by .

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

If it is clear which operation is used, then often only is written instead of .
(Authors: Burkhardt/Höllig/Hörner)