I am reading a book about Representation Theory of Finite Groups and in the book it gives:

$End(W) \cong M_n(\mathbb{C})$

$GL(V) \cong GL_n(\mathbb{C})$

$Hom(V,W) \cong M_{mn}(\mathbb{C})$

$V$ and $W$ are Vector Spaces and $Hom(V,W) = \{A:V\rightarrow W | A$ is a linear map$\}$, $End(V) = Hom(V,V)$ and $GL(V) = \{A \in End(V)|A$ is invertible$\}$

I understand what these sets are, but in the above relations I was unsure whether they are being considered as groups (With group operation composition of maps/multiplication of matrices) or as vector spaces (with addition of maps/matrices and scalar multiplication by scalars in $\mathbb{C}$).

## Best Answer

$\operatorname{End}(W) \cong M_n(\mathbb{C})$ as $\mathbb{C}$-algebras (i.e. as rings as well as $\mathbb{C}$-vector spaces), $\operatorname{GL}(V) \cong \operatorname{GL}_n(\mathbb{C})$ as groups, and $\operatorname{Hom}(V,W) \cong M_{mn}(\mathbb{C})$ as $\mathbb{C}$-vector spaces.

EDITNote that $\operatorname{End}(W)$ is not a group wrt composition (as it contains non-invertible elements) and $\operatorname{GL(V)}$ is not a vector space (as it doesn't contain an additive zero element, for example). So there is no way to read all three $\cong$ signs uniformly as "isomorphic as vector spaces" nor uniformly as "isomorphic as multiplicative groups".