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# prelim2 - Spring 2010 MATH 4320 Prelim 2 Instructor Yuri...

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Spring 2010 MATH 4320: Prelim 2 Instructor: Yuri Berest The exam is due Wednesday, April 14 . Please write clearly and concisely. Problem 1. (15 points) If H 1 and H 2 are two groups, define their direct product H 1 × H 2 to be the set of ordered pairs { ( x 1 , x 2 ) : x 1 H 1 , x 2 H 2 } equipped with the operation ( x 1 , x 2 ) · ( y 1 , y 2 ) = ( x 1 y 1 , x 2 y 2 ) . ( a ) Show that the natural maps π 1 : H 1 × H 2 H 1 , ( x 1 , x 2 ) x 1 , and π 2 : H 1 × H 2 H 2 , ( x 1 , x 2 ) x 2 , are surjective group homomorphisms. What are the kernels of π 1 and π 2 ? ( b ) Prove the following property of ( H 1 × H 2 , π 1 , π 2 ) : given any group G together with two homomorphisms f 1 : G H 1 and f 2 : G H 2 , there is a unique homomorphism ϕ : G H 1 × H 2 such that f 1 = π 1 ϕ and f 2 = π 2 ϕ . Problem 2. (20 points) Let N and H be subgroups of a group G with N normal. Let Ad g : G G , x gxg - 1 , be conjugation by an element g G . ( a ) Show that the assignment h Ad h defines a group homomorphism f : H Aut( N ).

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prelim2 - Spring 2010 MATH 4320 Prelim 2 Instructor Yuri...

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