Algebra 1 HW 3

Algebra 1 HW 3 - (Hint for(a If G/Z G = ° gZ G ± ±rst...

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Homework Assignment 3 Math 235 (Fall 2011) Due: Oct 12 (Wednesday) at the beginning of class (1) Let ψ : C × R × given by ψ ( z )= | z | ,where | x + yi | = ° x 2 + y 2 . (a) Show that ψ is a homomorphism. (b) Calculate ker( ψ )andim( ψ ). (2) Let φ : G H be a homomorphism and let g G have ±nite order. (a) Show that the order of φ ( g )d iv idestheordero f g . (b) If φ is an isomorphism, show that | φ ( g ) | = | g | . (3) Suppose that G is a group and G/Z ( G )iscyc l
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Unformatted text preview: (Hint for (a): If G/Z ( G ) = ° gZ ( G ) ± , ±rst show G = { g n z : z ∈ Z ( G ) , n ∈ Z } . ) (4) Show that a subgroup is normal if and only if it is the kernel of a homomorphism. (5) Let G be a group. Show that G/Z ( G ) ∼ = Inn( G ). Department of Mathematics, McGill University, Montreal, QC, H3A 2K6 Canada E-mail address : [email protected] 1...
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