Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Trivial group
Summary
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: or depending on the context. If the group operation is denoted then it is defined by
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 is distinct from the empty set, which has no elements, hence lacks an identity element, and so cannot be a group.
Given any group the group consisting of only the identity element is a subgroup of and, being the trivial group, is called the of
The term, when referred to " has no nontrivial proper subgroups" refers to the only subgroups of being the trivial group and the group itself.
The trivial group is cyclic of order ; as such it may be denoted or If the group operation is called addition, the trivial group is usually denoted by 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
The trivial group serves as the zero object in the , meaning it is both an initial object and a terminal object.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.