2. Consider the group (R, +) and the subgroup H = (2nx |n€ Z}. %3! [cos(0) - sin(@) sin(0) cos(0) Let G denote the group of matrices of the form where O is any real number and the group operation is matrix multiplication. Prove that R/H is isomorphic to G by first giving an explicit formula for a map o : R/H → G, and then checking that your map is well-defined, bijeetive, and respects the group operation.

2. Consider the group (R, +) and the subgroup H = (2nx |n€ Z}. %3! [cos(0) - sin(@) sin(0) cos(0) Let G denote the group of matrices of the form where O is any real number and the group operation is matrix multiplication. Prove that R/H is isomorphic to G by first giving an explicit formula for a map o : R/H → G, and then checking that your map is well-defined, bijeetive, and respects the group operation.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 5E

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