Projective general linear group: Difference between revisions

Revision as of 03:53, 27 August 2007

This term associates to every field, a corresponding group property. In other words, given a field, every group either has the property with respect to that field or does not have the property with respect to that field

This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

Definition

In terms of dimension

Let $n$ be a natural number and $k$ be a field. The projective general linear group of order $n$ over $k$ , denoted $P G L (n, k)$ is defined in the following equivalent ways:

It is the group of automorphisms of projective space of dimension $n - 1$ , that arise from linear automorphisms of the vector space of dimension $n$ .
It is the quotient of $G L (n, k)$ by its center, viz the group of scalar multiplies of the identity (isomorphic to the group $k^{*}$ )

In terms of vector spaces

Let $V$ be a vector space over a field $k$ . The projective general linear group of $V$ , denoted $P G L (V)$ , is defined as the inner automorphism group of $G L (V)$ , viz the quotient of $G L (V)$ by its center, which is the group of scalar multiples of the identity transformation.

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 * It is the group of automorphisms of [[projective space]] of dimension <math>n-1</math>, that arise from linear automorphisms of the vector space of dimension <math>n</math>.
-* It is the quotient of <math>GL(n,k)</math> by its center, viz the group of scalar multiplies of the identity (isomorphic to the group <mathk^*</math>)
+* It is the quotient of <math>GL(n,k)</math> by its center, viz the group of scalar multiplies of the identity (isomorphic to the group <math>k^*</math>)
 ===In terms of vector spaces===
 Let <math>V</math> be a [[vector space]] over a field <math>k</math>. The projective general linear group of <math>V</math>, denoted <math>PGL(V)</math>, is defined as the [[inner automorphism group]] of <math>GL(V)</math>, viz the quotient of <math>GL(V)</math> by its center, which is the group of scalar multiples of the identity transformation.