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Group Homomorphism

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Suppose that you are given two groups H and K. A group homomorphism is a function \phi: H \rightarrow K which respects the group operations. This means that,

\phi(ab)=\phi(a)\phi(b) for all a, b \in H

Examples of Group Homorphisms:

1. Let (Z,+) denote the group of integers with addition and let (2Z,+) denote the group of even integers with addition.

Consider the function \phi: Z \rightarrow 2Z given as,

\phi(a)=2a for all a \in Z

This is clearly a group homomorphism because,

\phi(a+b)=2(a+b)=2a+2b=\phi(a)+\phi(b) for all a \in Z

In fact, it is a group isomorphism because the function \phi is bijective. This means that the two groups are different copies of the same underlying structure.

2. Let S_n denote the group of permutations on ‘n’ symbols. Consider the function, \phi: S_n \rightarrow \{+1,-1\} which takes an element of S_n to its sign.

This homomorphism is called the sign homomorphism and is one of the classic examples of a group homomorphism. It takes an element with an even number of cycles to +1 and those with an odd number of cycles to -1.

Why study group homomorphims?

We study group homomorphisms because mapping a complicated group onto another much simpler group while preserving the structure of the binary operation allows us to focus on some particular aspect of the group structure.

For example, studying the sign homomorphism in the above example allows us to focus on the sign of a permutation. The homomorphism “forgets” all other aspects of the group structure and enables us to focus on the property we wish to understand.

What is a group isomorphism?

A group isomorphism is a group homomorphism that is both injective and surjective. If there is an isomorphism between two groups we say that the groups are isomorphic. This means that the two groups are essentially the same. They are copies of each other with the symbols “relabelled”.

As seen above, Z and 2Z are isomorphic considered as groups under addition.

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