PGL–PSL comparison

Statement

Let $n\ge2$ and let $k$ be a field. Determinant induces a well-defined surjective homomorphism

\overline{\det}:\operatorname{PGL}_n(k)\longrightarrow k^\times/(k^\times)^n,\qquad [A]\longmapsto [\det A],

whose kernel is the embedded $\operatorname{PSL}_n(k)$ . Hence there is a short exact sequence of abstract groups

1\longrightarrow\operatorname{PSL}_n(k) \longrightarrow\operatorname{PGL}_n(k) \xrightarrow{\ \overline{\det}\ } k^\times/(k^\times)^n \longrightarrow1.

Why the determinant descends

Replacing $A$ by another representative $\lambda A$ of its projective class multiplies the determinant by $\lambda^n$ , so its class modulo $n$ -th powers is unchanged. Surjectivity follows from

\det\operatorname{diag}(a,1,\ldots,1)=a.

If $\det A=\lambda^n$ , then $\lambda^{-1}A\in\operatorname{SL}_n(k)$ and represents the same projective class. This proves the kernel statement.

Important fields

For $k=\mathbb C$ , every nonzero complex number is an $n$ -th power, so

\operatorname{PGL}_n(\mathbb C)=\operatorname{PSL}_n(\mathbb C).

For $k=\mathbb R$ , the quotient is trivial when $n$ is odd and has two elements when $n$ is even. Therefore

\operatorname{PGL}_n(\mathbb R)=\operatorname{PSL}_n(\mathbb R) \quad\text{for odd }n,

whereas $\operatorname{PSL}_n(\mathbb R)$ has index $2$ in $\operatorname{PGL}_n(\mathbb R)$ for even $n$ . In particular, $\operatorname{PSL}_2(\mathbb R)\ne\operatorname{PGL}_2(\mathbb R)$ .

For a finite field $\mathbb F_q$ , the quotient has order $\gcd(n,q-1)$ , since $\mathbb F_q^\times$ is cyclic. Thus equality is controlled by arithmetic in the ground field, not by notation.

Scope warning

This exact sequence concerns groups of $k$ -valued matrices modulo scalar matrices. It should not be confused with an exact sequence of group schemes followed blindly by $k$ -points: taking rational points of a quotient need not be right-exact.

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: central quotients and classical groups.
Roger W. Carter, Simple Groups of Lie Type, Wiley, 1972. Publisher record. Relevant: Chapter 1, projective linear groups over finite fields.