Trivial group

Inmathematics,atrivial grouporzero groupis agroupconsisting of a single element. All such groups areisomorphic,so one often speaks ofthetrivial group. The single element of the trivial group is theidentity elementand so it is usually denoted as such:$0,1,$or$e$depending on the context. If the group operation is denoted$\,\cdot \,$then it is defined by$e\cdot e=e.$

The similarly definedtrivial monoidis also a group since its only element is its own inverse, and is hence the same as the trivial group.

The trivial group is distinct from theempty set,which has no elements, hence lacks an identity element, and so cannot be a group.

Definitions

Given any group$G,$the group consisting of only the identity element is asubgroupof$G,$and, being the trivial group, is called thetrivial subgroupof$G.$

The term, when referred to "$G$has no nontrivial proper subgroups "refers to the only subgroups of$G$being the trivial group$\{e\}$and the group$G$itself.

Properties

The trivial group iscyclicof order$1$;as such it may be denoted$\mathrm {Z} _{1}$or$\mathrm {C} _{1}.$If the group operation is called addition, the trivial group is usually denoted by$0.$If the group operation is called multiplication then 1 can be a notation for the trivial group. Combining these leads to thetrivial ringin which the addition and multiplication operations are identical and$0=1.$

The trivial group serves as thezero objectin thecategory of groups,meaning it is both aninitial objectand aterminal object.

The trivial group can be made a (bi-)ordered groupby equipping it with the trivialnon-strict order$\,\leq.$

See also

• Zero object (algebra)– Algebraic structure with only one element
• List of small groups

References

• Rowland, Todd & Weisstein, Eric W."Trivial Group".MathWorld.