1. Prove that the group C* (= multiplicative group of non-zero comp...

1. Prove that the group C* (= multiplicative group of non-zero complex numbers) is isomorphic to the direct product (R*₊) x (R/Z) of the multiplicative group R*₊ of positive real numbers and the quotient group of R mod Z (= additive group of real numbers factorized by the subgroup of integers).

2. Prove that a subgroup of index 2 is normal. (Index of a subgroup is the number of cossets -if it is finite.) Hint: Given an irregularly shaped bottle partially filled with water, how to find out if the contents are more than half-a-bottle or less?

3. Describe up to isomorphism all homomorphic images of: (a) Z₁₀₀₁, (b) Z*₃₂.

4. In the dihedral group Dₙ (of symmetries of a regular n-gon), describe all classes of conjugate elements, and use the result to describe all normal subgroups H in Dₙ. For each of them, construct the quotient group Dₙ/H, and prove that it is isomorphic to either a cyclic or dihedral, and find out to which cyclic or to which dihedral.

5. (a) Find all normal subgroups H in S₄. Hint: Project H by the homomorphism S₄ -> S₃ (and use the fact that S₃ is dihedral.)
(b) Show that every homomorphic image of S₄ is isomorphic to one of the groups S₄, S₃, S₂, or S₁.

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