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Question: problem 3 i let h be a subgroup of a...

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Problem 3. i) Let H be a subgroup of a group G. Let K be a conjugate of H. Prove that H and K are isomorphioc (3pts). Hint: write down what it means for K being a conjugate of H, then use it to write down the most natural/obvious way to obtain an element of K from a given element h of H MATH 315 - ASSIGNMENT 2 (ii) Let f : G → G be a surjective homomorphism (also called epimorphism) from to another group G. Prove that f(Z(G))-Z(G). (3pts) Consider the determinant homomorphism mentioned in the lecture Prove that the homomorphism det is surjective and describe the image of Z(GL(2,R)) (3pts). Hint: (ii) This illustrates that we dont necessarily have equality between f(Z(G)) and Z(G1) in part (ii) det: GL(2,R) R. use the description of Z(GL(2, R)) in the note Tutorial2.

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