Group homomorphisms and isomorphisms

Say we have a group \(G\) with elements \(\require{physics} \set{a, b, \cdots}\) and group multiplication \(\cdot\). Say we also have another group \(G'\) with elements \(\set{a', b', \cdots}\) and group multiplication \(\cdot'\). A group homomorphism is a function \[\begin{equation} h: G \rightarrow G': g \mapsto g' = h(g) \end{equation}\] such that \[\begin{equation} \forall a, b \in G: h(a \cdot b) = h(a) \cdot' h(b) \thinspace . \end{equation}\] From this definition we can show that the identity \(e\) of \(G\) is mapped onto the identity \(e'\) of \(G'\) and that \[\begin{equation} \forall a \in G: h(a^{-1}) = h(a)^{-1} \end{equation}\] such that we can say that the function \(h\) (the relation between the two groups) is compatible with the group structure.

Equivalently, we can write for a group homomorphism \(h\): \[\begin{align} h: G \rightarrow G': &\forall a, b, c \in G: \\ &a \cdot b = c \Rightarrow h(a) \cdot' h(b) = h(c) \thinspace . \end{align}\]

A special type of group homomorphism is a group endomorphism. This is a group homomorphism from a set \(G\) to itself: \[\begin{equation} h: G \rightarrow G \end{equation}\]

Another special group homomorphism is a group isomorphism. It is a group homomorphism that is bijective.

Let us consider a subset \(C\) of the matrices with real entries (\(x, y \in \mathbb{R}\)), consisting of matrices of the form \[\begin{equation} \begin{pmatrix} x & y \\ -y & x \end{pmatrix} \end{equation}\] Then, defining \(f\) as the field isomorphism \(f: C \rightarrow \mathbb{C}\); \[\begin{equation} \begin{pmatrix} x & y \\ -y & x \end{pmatrix} \mapsto x + yi \thinspace , \end{equation}\] it can be seen that the field \((C, +, \cdot)\) is isomorphic to the field \((\mathbb{C}, +, \cdot)\).

We have seen that \(\text{Aut}(V)\) is the automorphism group of \(V\). By introducing a basis in \(V\), we can represent these invertible linear operators as invertible linear matrices and we can say that \(\text{Aut}(V)\) is isomorphic (one-to-one correspondence) to \(\text{GL}(n)\).