Projective special linear group

Definition

Let $n\ge2$ and let $k$ be a field. The projective special linear group is

\operatorname{PSL}_n(k) :=\operatorname{SL}_n(k)/Z(\operatorname{SL}_n(k)).

The center consists exactly of scalar matrices

Z(\operatorname{SL}_n(k)) =\{\lambda I_n:\lambda\in k^\times,\ \lambda^n=1\}.

Projective realization

The inclusion $\operatorname{SL}_n(k)\hookrightarrow\operatorname{GL}_n(k)$ followed by the quotient to the projective general linear group has kernel $Z(\operatorname{SL}_n(k))$ . It therefore identifies $\operatorname{PSL}_n(k)$ with a normal subgroup of $\operatorname{PGL}_n(k)$ , acting faithfully on $\mathbb P^{n-1}(k)$ .

This is a definition of the abstract group of $k$ -points. The central quotient of the corresponding algebraic group and its $k$ -points require care over non-algebraically closed fields; the present page makes no assertion that taking $k$ -points commutes with every quotient construction.

Comparison with

\operatorname{PGL}

Over an algebraically closed field, every nonzero scalar has an $n$ -th root, and $\operatorname{PSL}_n(k)=\operatorname{PGL}_n(k)$ as subgroups of projective transformations. Over a general field they may differ; the precise quotient is $k^\times/(k^\times)^n$ .

References

James S. Milne, Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press, 2017. Author-maintained text. Relevant: $\operatorname{SL}_n$ , centers, and central quotients.
Jean-Pierre Serre, Linear Representations of Finite Groups, Springer, 1977. Publisher record. Relevant: projective linear groups as standard finite-group examples.