PGL(2,c) is isomorphic to PSL(2,c)

Question

Definition: Projective special linear group

$PSL\left ( n,\mathbb{F} \right )=\frac{SL\left ( n,\mathbb{F} \right )} {\left ( Z\left ( GL\left ( n,\mathbb{F} \right ) \right )\cap SL\left ( n,\mathbb{F} \right ) \right )}$

Proof:

Let $A \in GL\left ( 2,\mathbb{C} \right )$ and let >$\lambda=\sqrt{det\left ( A \right )^{-1}}\in\mathbb{C}$

Then, $det\left ( \left ( \lambda I \right )A \right )=det\left ( \lambda I \right )det\left ( A \right )=\lambda^{2}det\left ( A \right )=1$

So, $\lambda A=SL\left ( 2,\mathbb{C} \right )$.

Since $A =\left ( \lambda^{-1} I \right )\left ( \lambda A \right )$

we have

$\mathbf{\left ( 1 \right )}$

$\mathbf{ GL\left ( 2,\mathbb{C} \right )=SL\left ( 2,\mathbb{C} \right )Z\left ( GL\left ( 2,\mathbb{C} \right ) \right )}$

and so,

$\mathbf{\left ( 2 \right )}$

$\mathbf{PGL\left ( 2,\mathbb{C} \right )=PSL\left ( 2,\mathbb{C} \right )}$

Why is every matrices in GL the product of some matrices in SL and in the center of the group GL? (1) and (2) in bold is not very clear to me. Thanks in advance.

The first one is because the center of $GL_n$ precisely consists of the scalar matrices. The second is because of one of the isomorphism theorems. — Tobias Kildetoft, May 05 '16 at 06:15
@TobiasKildetoft I don't quite understand how "factoring out" Z from SL gives PSL. Could you explain? — Mathematicing, May 05 '16 at 06:42
I am not sure which part you are asking about. By definition $PSL$ is the quotient of $SL$ by its center. — Tobias Kildetoft, May 05 '16 at 06:43
Point (2) is the question I am asking. What I meant was how does "factoring out" SL(2,C)Z gives SL/Z?@TobiasKildetoft — Mathematicing, May 05 '16 at 06:44
As I said, that is one of the isomorphism theorems $HN/N\cong H/(H\cap N)$. — Tobias Kildetoft, May 05 '16 at 06:46
How silly of me. By the second Isomorphism. @TobiasKildetoft Thanks! — Mathematicing, May 05 '16 at 06:47

PGL(2,c) is isomorphic to PSL(2,c)

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