# Are End(V), GL(V) and Hom(V,W) Groups or Vector spaces?

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 $$\mathbb{C}$$).

• Which book are you referring to? Oct 26, 2021 at 20:17
• "Benjamin Steinberg - Representation Theory of Finite Groups: An Introductory Approach" Oct 26, 2021 at 20:19

$$\operatorname{End}(W) \cong M_n(\mathbb{C})$$ 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.
Note 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".
• @123123 No, that's not possible. $\operatorname{End}(W)$ is not a group, and $\operatorname{GL}(V)$ is not a vector space, so no way to read all $\cong$ signs as "isomorphic as gropus" nor as "isomorphic as vector spaces". Oct 26, 2021 at 20:22
• $End(V)$ is not a group under multiplication. As a vector space, or as a ring, it is always a group under addition. Oct 26, 2021 at 20:51