The 5th roots of unity in the complex plane under multiplication form a group of order 5. Each non-identity element by itself is a generator for the whole group.
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set.
In the study of differential equations, and commonly those occurring in physics, one has the idea of a set of infinitesimal displacements that can be extended to obtain a manifold, or at least, a local part of it, by means of integration. The general concept is of using the exponential map to take the vectors in the tangent space and extend them, as geodesics, to an open set surrounding the tangent point. In this case, it is not unusual to call the elements of the tangent space the generators of the manifold. When the manifold possesses some sort of symmetry, there is also the related notion of a charge or current, which is sometimes also called the generator, although, strictly speaking, charges are not elements of the tangent space.
Elements of the Lie algebra to a Lie group are sometimes referred to as "generators of the group," especially by physicists.[1] The Lie algebra can be thought of as the infinitesimal vectors generating the group, at least locally, by means of the exponential map, but the Lie algebra does not form a generating set in the strict sense.[2]
