What is the size of $\operatorname{PSL}_2(K)$?

Question

I'm trying to figure out what the size of the projective special linear group $$\operatorname{PSL}_n(K) := \operatorname{SL}_n(K)/Z(\operatorname{SL}_n(K))$$ is, when the field $K$ has $q$ elements and in case $n=2$.

I was able to figure out the size of the special linear group $$|\operatorname{SL}_n(K)| = q^{\frac{1}{2}n(n-1)}\prod_{i=2}^n(q^i - 1)$$ which is correct acording to the solution. So I tried to get the size of the center, and the solution states $$|Z(\operatorname{SL}_2(K))| =\begin{cases}1,\quad \text{ if $q$ is even} \\ 2, \quad\text{ if $q$ is odd}\end{cases} $$

Question: The first case seems clear so far, but I'm having a hard time to understand the second case. What is the second element in the center?

If $q$ is even, you only have $I_2$ in the center (because $+1=-1$ if $q$ is even), but if $q$ is odd, you have $I_2$ and $-I_2$. — TheSilverDoe, May 25 '21 at 18:52
Thanks, but how is that connected to the odd/even attribute? — Algebruh, May 25 '21 at 18:56
If $q$ is even, then you are in a field of characteristic $2$, so $1=-1$, so $I_2=-I_2$. — TheSilverDoe, May 25 '21 at 18:56

Sam Freedman · Accepted Answer · 2021-05-25T19:04:47.507

The center of $\text{SL}(2, K)$ is isomorphic to $\{\lambda \in K : \lambda^2 = 1\}$ (see this question). This follows more generally from the fact that the center of $\text{GL}(2, K)$ consists of diagonal matrices (see here).

The solutions to $\lambda^2 = 1$ in $K$ are $1$ and $-1$, yet when the characteristic of $K$ is $2$ these elements are the same. When $K$ is even there is therefore only one element in the center---the identity $I$. But when $K$ is odd the characteristic is greater than 2, so $I$ and $-I$ are two distinct elements of the center.

score 0 · Answer 2 · answered May 25 '21 at 18:53

0

$$\begin{pmatrix} -1 \\& -1 \end{pmatrix}$$

answered May 25 '21 at 18:53

D_S

33,891

Thank you! But how is that connected to the even/uneven thing? – Algebruh May 25 '21 at 18:55
If $q$ is even, then $-1$ is the same thing as $1$ and this is just the identity element. – D_S May 25 '21 at 19:07

What is the size of $\operatorname{PSL}_2(K)$?

2 Answers2