I have three questions about algebraic groups.
Let $D$ be a linear algebraic group. Then the following are equivalent:
$D$ is diagonalizable.
$\mathop{Hom}(D,\mathbb{C}^*)$ is finitely generated with an isomorphism of coordinate rings $\mathbb{C}[D]\cong \mathbb{C}[\mathop{Hom}(D,\mathbb{C}^*)]$.
Every rational representation of $D$ is isomorphic to a direct sum of $1$-dimensional representations.
$D$ is isomorphic to $(\mathbb{C}^*)^r\times A$, for some finite abelian group $A$.
I'm fine with #1-3, but I had to pause after reading #4 because I am now wondering why and how does this torsion part arise (it may be because I do not know any algebraic groups other than the ones that can (naturally) be embedded into $GL(n,\mathbb{C})$).
Example: Take $D$ to be a diagonal group with character group $\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$. Then the coordinate ring of $D$ is $\mathbb{C}[x,x^{-1},y]/\langle y^2-1\rangle$ with $D\cong \mathbb{C}^*\times \mu_2$.
$\mathbf{Question \;1}$: Is there a way to write $D\cong \mathbb{C}^*\times \mu_2$ in a single matrix form, i.e., embed $D$ into $GL(n,\mathbb{F})$? Or must it be separately embedded into a product of $GL_n$'s like $GL(1,\mathbb{C})\times GL(1,\mathbb{F}_2)$?
Suppose our above $D\cong \mathbb{C}^*\times \mu_2$ acts on $\mathbb{C}^3$ by $ (x,y).(a,b,c)=(xa,x^{-1}b,yc). $
$\mathbf{Question\; 2}$: Then do we have $ (x,y)^2.(a,b,c)=(x^2a,x^{-2}b,c) \mbox{ with } \mathbb{C}[a,b,c]^D=\mathbb{C}[ab,c^2]? $
$\bf{Question \; 3}$: Can you give me an example of an algebraic group which cannot be embedded into a product of $GL_n$'s?
Thank you.