Unformatted text preview: special linear group , consists of the matrices with determinant 1. The projective versions of these groups are obtained by taking the quotient with their respective centers (scalars of the identity matrix). That is, PGL 2 ( R ) = GL 2 ( R ) / h cI i , PGL + 2 ( R ) = GL + 2 ( R ) / h cI i , PSL 2 ( R ) = SL 2 ( R ) / {± I } . (a) Prove that the center of the general linear group, Z (GL 2 ( R )) = h cI  c ∈ R × i , is a subset (and hence a a subgroup) of GL + 2 ( R ). (b) Prove that PGL + 2 ( R ) is isomorphic to PSL 2 ( R ). [ Hint : Use the ﬁrst isomorphism theorem – if ϕ : G → H is a homomorphism, then G/ ker ϕ ∼ = im ϕ .] MthSc 481  Topics in Geometry and Topology  Spring 2012  M. Macauley...
View
Full Document
 Spring '12
 Staff
 Geometry, Topology, General linear group, M. Macauley

Click to edit the document details