ldelong said:Group G must be communitative for it to be abelian
A homomorphism is a mathematical function that preserves the structure of a group or algebraic structure, meaning that the operation being performed on the elements of the structure is preserved by the function.
A homomorphism is abelian if the order in which the elements are multiplied does not affect the outcome of the operation. This means that the function commutes, or that the elements can be rearranged without changing the result.
Homomorphisms help in understanding algebraic structures by showing the relationships between different structures and how they are related through their underlying operations. They also help in identifying isomorphisms, which are structures that are essentially the same, even though they may have different representations.
Some common examples of homomorphisms include the logarithmic function, which preserves multiplication, and the absolute value function, which preserves addition. Another example is the determinant function in linear algebra, which preserves the matrix multiplication operation.
Homomorphisms have many practical applications, especially in cryptography and coding theory. They can be used to encrypt data by transforming it into a different mathematical structure, making it difficult to decipher without the proper key. They can also be used to detect errors in data transmission, as any changes to the data will result in a different value after the homomorphism is applied.