Trivial group

In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are

isomorphic

, so one often speaks of the trivial group. The single element of the trivial group is the identity element and 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 defined is also a group since its only element is its own inverse, and is hence the same as the trivial group.

The trivial group should not be confused with the

empty set

, which has no elements, and lacking an identity element, cannot be a group.

Definitions

Given any group $G,$ the group consisting of only the identity element is a subgroup of $G,$ and, being the trivial group, is called the of $G.$

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

Properties

The trivial group is

cyclic

of order $1$; as such it may be denoted $Z_1$ or $C_1.$ 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 the trivial ring in which the addition and multiplication operations are identical and $0 = 1.$

The trivial group serves as the zero object in the category of groups, meaning it is both an initial object and a terminal object.

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

See also

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References

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{{DEFAULTSORT:Trivial Group
Finite groups