Triviaaliryhmä

Triviaaliryhmä is the smallest possible group in abstract algebra. It consists of only one element, usually denoted by 'e' or '0', which acts as the identity element under the group operation. The group operation is defined such that any element combined with the identity element results in the identity element itself. For example, if the operation is denoted by '*', then e * e = e.

Because there is only one element, the group axioms are trivially satisfied. The closure property holds as e * e = e, which is an element of the group. The associative property holds because there are no choices to be made. The identity element is the element e itself. The inverse of the identity element is also the identity element, since e * e = e.

The trivial group is unique up to isomorphism. This means that any group with only one element is structurally the same as any other group with only one element. It is often used as a base case or a simple example when introducing group theory concepts. The trivial group can be considered as the zero object in the category of groups.