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s12_mthsc481_hw06

# s12_mthsc481_hw06 - special linear group consists of the...

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Homework 6 | Due March 15 (Thursday) 1 Read : Stahl, Chapters 9.1, 9.2. 1. Prove that the rotations R 0 and R 0 , - α are conjugate in Isom( E 2 ). 2. Prove that the glide reﬂections γ AB and γ CD are conjugate in Isom( E 2 ) if and only if d ( A,B ) = d ( C,D ). 3. Let f = γ AB , g = γ AC , where A , B , and C are distinct points of E 2 . Describe the Euclidean isometry fgf - 1 in standard form (i.e., as a translation, rotation, reﬂection, or glide-relection). 4. Find the image of the points i , 1+ i , and - 3+4 i when subjected to the following isometries, where 0 = (0 , 0) and A = (3 , 0). (i) I A, 2 (ii) f ( z ) = 2 z - 1 z + 2 . 5. Express the following compositions as M¨ obius transformations, where 0 = (0 , 0) and A = (3 , 0). (i) I A, 2 I 0 , 3 (ii) 2( - ¯ z ) - 1 ( - ¯ z ) + 2 I 0 , 3 (iii) 2( - ¯ z ) - 1 ( - ¯ z ) + 2 2 z - 1 z + 2 . 6. Consider the following subgroups of GL 2 ( R ), the degree-2 general linear group : GL + 2 ( R ) consists the matrices with positive determinant, and SL 2 ( R ), called the
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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...
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