Trivial group
In group theory , the trivial group is a group whose carrier set contains exactly one element . The trivial group is clearly defined except for isomorphism . Each group contains the trivial group as a subgroup .
definition
The trivial group is a group that consists of the one-element set and is provided with the only possible group operation
- .
The element is thus the neutral element of the group.
Examples
All trivial groups are isomorphic to one another . Examples of trivial groups are:
- the cyclic group of the degree
- the alternating group of degree
- the symmetric group of a one-element set
properties
- Since the group operation is commutative, the trivial group is an Abelian group .
- The only subgroup of the trivial group is the trivial group itself.
- The trivial group of the empty set generated : . Here, according to the usual convention , the empty product results in the neutral element.
- Each group contains the trivial group and itself as the ( trivial ) normal divisor . The trivial group is therefore mostly not seen as a simple group .
- In the category of groups Grp , the trivial group functions as a null object .
See also
literature
- Rainer Schulze-Pillot: Introduction to Algebra and Number Theory . Springer, 2008, ISBN 3-540-79570-7 .
- Jürgen Wolfart: Introduction to number theory and algebra . Springer, 2010, ISBN 3-8348-9833-3 .
Web links
- Todd Rowland, Eric W. Weisstein : Trivial Group . In: MathWorld (English).