Projective special linear group

Example

Let $F$ be a field and let $n \ge 2$ be an integer. The projective special linear group is the quotient group $$PSL_n(F) = SL_n(F) / Z,$$ where $SL_n(F)$ is the special linear group and $Z$ is its centre. The centre $Z$ consists of the scalar matrices $\lambda I_n$ with $\lambda^n = 1$. For a finite field $\F_q$ with $q$ elements, write $PSL_n(q)$.

Take $n = 2$ and $q = 7$. The order of $SL_2(7)$ is $q(q^2 - 1) = 7 \cdot 48 = 336$. The scalars $\lambda I_2$ with $\lambda^2 = 1$ are $\pm I_2$, so $\abs{Z} = 2$. Hence $PSL_2(7)$ has order $168$. It is a simple group, and it is the second smallest non-abelian simple group, after the alternating group $A_5$ of order $60$.