Trivial and nontrivial are terms used in group theory to describe the complexity or simplicity of a group. In this context, a group refers to a set of elements along with an operation that combines any two elements to produce a third element. The operation must satisfy certain properties, such as associativity, identity, and inverse elements.
The trivial group is the simplest possible group, consisting of just one element. This single element is usually denoted as the identity element or simply the identity. The operation in the trivial group is such that combining the identity element with itself or any other element always results in the identity element. In other words, the trivial group is closed under the group operation, and every element in the group is its own inverse.
One way to visualize the trivial group is to think of a group of people where there is only one person. In this case, there are no interactions or operations to perform between individuals since there is only one person present. It is a simple and uninteresting scenario, hence the term “trivial.”
On the other hand, nontrivial groups are more complex and interesting. They have more than one element, and the group operation involves combining these elements in different ways. Nontrivial groups can have various properties and structures, which can be studied and analyzed in group theory.
To illustrate the concept of nontrivial groups, let’s consider the group of integers under addition. In this group, the elements are all the integers, and the group operation is ordinary addition. The identity element is 0, and for every integer, there is an inverse element such that adding the integer and its inverse gives the identity element.
The group of integers under addition is nontrivial because it contains an infinite number of elements, and the group operation has interesting properties. For example, adding two positive integers always results in a positive integer, while adding a positive and a negative integer can yield a positive, negative, or zero value. These properties of the group structure make it nontrivial and worth studying.
The trivial group is the simplest possible group, consisting of just one element, while nontrivial groups have more than one element and exhibit more complex structures and properties. The trivial group is uninteresting from a group theory perspective, while nontrivial groups provide a rich field for exploration and analysis.