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 ${\displaystyle (\{e\},*)}$ ${\displaystyle \{e\}}$

${\displaystyle e*e=e}$.

The element is thus the neutral element of the group. ${\displaystyle e}$

Examples

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

• the cyclic group of the degree ${\displaystyle C_{1}}$ ${\displaystyle 1}$
• the alternating group of degree ${\displaystyle A_{2}}$ ${\displaystyle 2}$
• the symmetric group of a one-element set ${\displaystyle S_{1}}$

properties

• Since the group operation is commutative, the trivial group is an Abelian group . ${\displaystyle \ast }$
• 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. ${\displaystyle \{e\}=\langle \emptyset \rangle }$
• 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

• Zero ring
• Zero vector space

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).