Group theory and group representation

Question

I am fairly new to group theory and representation. I am currently looking at faithful representations. I am not quite sure what is the "use" of a faithful representation. I cannot find any "easy to understand" literature on this too. To be specific, say I am finding a representation which is faithful of the group $\mathbb{Z_2}$ x $\mathbb{Z_2}$ in $GL_2(\mathbb{Z})$. So I looked at the generator of the group, say $x$ and $y$, I send $x \rightarrow \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ and $y \rightarrow \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ and all the other elements based on this map. My question is so what if we have this? Why does this at all have to be injective? I apologize if this is a very trivial question.

Answer 1

A partial answer is the image of a faithful representation $\mathbb{V}$ of a group (endowed with the group operation of composition, or in a basis, matrix multiplication) is isomorphic to the group itself, and hence realizes your group as a subgroup of $GL(\mathbb{V})$ (or given a basis, as an explicit matrix group), whereas the image of a nonfaithful (faithless?) representation contains less information than the group itself.

In your example, your representation is faithful, and the group $\mathbb{Z}_2 \times \mathbb{Z}_2$ is isomorphic to the matrix group $$ \left\{ \left(\begin{array}{cc} 1 & 0\\0 & 1\end{array}\right), \left(\begin{array}{cc} 1 & 0\\0 & -1\end{array}\right), \left(\begin{array}{cc}-1 & 0\\0 & 1\end{array}\right), \left(\begin{array}{cc}-1 & 0\\0 & -1\end{array}\right) \right\} . $$

However, the representation characterized by $$ x \mapsto \left(\begin{array}{cc} 1 & 0\\0 & -1\end{array}\right), \qquad y \mapsto \left(\begin{array}{cc} 1 & 0\\0 & 1\end{array}\right), \qquad $$ is not faithful, and its image is $$ \left\{ \left(\begin{array}{cc} 1 & 0\\0 & 1\end{array}\right), \left(\begin{array}{cc} 1 & 0\\0 & -1\end{array}\right) \right\} , $$ which is isomorphic to $\mathbb{Z}_2$, and not to $\mathbb{Z}_2 \times \mathbb{Z}_2$.

Answer 2

A faithful representation gives you, for example, a way to realize your group as a subgroup of $GL_n$ for some $n$. Sometimes you may have a group that's been defined in some abstract way and its set of elements are hard to write down explicitly. Then if you can find a faithful representation and find out what its image is, you can 'represent' your group as some matrices which you can write down and play with.