condmatscott said:I start with the group GL(2,N), where N is prime. I want to break these elements into N classes. One way to do this would be to find a homomorphism to Z_N, does such a homomorphism exist for general N? What is it? Is there another way to break the group into classes without using a homomorphism?
A homomorphism is a mathematical function that preserves the structure and operations of a group. In other words, if we apply the function to two elements of the group, the result will still be an element of the group.
GL(2,N) is the set of all invertible 2x2 matrices with entries in the set of natural numbers (positive integers).
Z_N is the set of integers modulo N, where N is a positive natural number. This means that the elements of Z_N are the remainders when dividing integers by N.
A homomorphism from GL(2,N) to Z_N is a function that maps each invertible 2x2 matrix to an element in Z_N. This mapping preserves the group structure, meaning that the operation of matrix multiplication in GL(2,N) is preserved in the resulting elements of Z_N.
Homomorphisms from GL(2,N) to Z_N are used in various fields of mathematics, including group theory, number theory, and cryptography. They also have applications in physics and engineering, such as in the study of symmetries and transformations.