Dihedral group of order 8. Let S be the plane: S = {(x, y)|x, y R} and consider

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Dihedral group of order 8. Let S be the plane: S = {(x, y)|x, y R} and consider f, g defined by f(x, y) = (-x, y) (reflection about the y-axis) and g(x, y) = (- y, x) (counterclockwise rotation by 90 degrees). Define G = {figj|i = 0,1; j=0,1,2,3} and let * be the composition operation. Then f2=g4 = e. Prove the following: g* f = f *g-1 (Find g-1 explicitly in an function form, and then show this identity by computing each side on (x, y).) Fill in the remaining identities in the form figj. Think geometrically! G is a group-make the composition table so that you show closure by writing the entries in the form figj. NOTE: To show closure, you must write each table entry in the form figj. To do this, you will be helped by applying the identities from part (b). G is nonabelian

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You are offering only 300 points for a very exhausting question. And next time, try to post a clearer (bigger?) picture.

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