Trivial group

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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 .


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.


All trivial groups are isomorphic to one another . Examples of trivial groups are:


  • 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


  • 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 .

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