A Mobius map is a function that maps points from the extended complex plane to itself. It is a fractional linear transformation of the form f(z) = (az + b)/(cz + d), where a, b, c, and d are complex numbers and ad - bc ≠ 0. These maps are also known as conformal maps, as they preserve angles and shapes.
A Simple group is a type of mathematical group that does not contain any non-trivial normal subgroups. In simpler terms, this means that the group cannot be broken down into smaller groups. Simple groups have many important applications in mathematics, including in the theory of Lie groups and in abstract algebra.
Mobius maps form a Simple group called the Mobius group, or the group of projective transformations. This group is defined as the set of all Mobius maps that can be composed together to form a new Mobius map. The identity element of this group is the identity function f(z) = z, and the inverse of any Mobius map is also a Mobius map, satisfying the properties of a group.
The Mobius group has several important properties, including closure, associativity, identity element, and inverse element. These properties make it a group, and a Simple group in particular. Additionally, Mobius maps in this group preserve angles and shapes, making it useful in geometric transformations.
Mobius maps and the Mobius group have various applications in mathematics, physics, and engineering. For example, they are used in the study of conformal mappings, which have applications in aerodynamics, fluid mechanics, and electromagnetic fields. They are also used in computer graphics and image processing to create transformations that preserve angles and shapes.