Projective group

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

in $n$ variables over a skew-field $K$

The group $\def\PGL{ {\rm PGL}}\PGL_n(K)$ of transformations of the $(n-1)$-dimensional projective space $P^{n-1}(K)$ induced by the linear transformations of $K^n$. There is a natural epimorphism $$P: {\rm GL}_n(K)\to \PGL_n(K),$$ with as kernel the group of homotheties (cf. Homothety) of $K^n$, which is isomorphic to the multiplicative group $Z^*$ of the centre $Z$ of $K$. The elements of $\PGL_n(K)$, called projective transformations, are the collineations (cf. Collineation) of $P^{n-1}(K)$. Along with $\PGL_n(K)$, which is also called the full projective group, one also considers the unimodular projective group $\def\PSL{ {\rm PSL}}\PSL_n(K)$, and, in general, groups of the form $P(G) \subset \PGL_n(K)$, where $G \subset {\rm GL}_n(K)$ is a linear group.

For $n\ge 2$ the group $\PSL_n(K)$ is simple, except for the two cases $n=2$ and $|K|=2$ or 3. If $K$ is the finite field of $q$ elements, then

$$|\PSL_n(K)| = (q-1,n)^{-1} q^{n(n-1)/2} (q^n-1)(q^{n-1}-1)\cdots (q^2-1).$$

For a brief resumé on the orders of the other finite classical groups, like ${\rm PSp}_n$, and their simplicity cf. e.g. [Ca].

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