hw5 - (3 points) 3. Let G be a group, and suppose there is...

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Math 4124 Wednesday, February 23 Fifth Homework Due 2:30 p.m., Monday March 14 1. Let G be a group of order 10 with a normal subgroup H of order 2. (a) Prove that H is contained in the center of G (hint: If 1 6 = x H and g G , then 1 6 = gxg - 1 H ). (b) Prove that G is abelian. (c) Let a G \ H . Prove that aH has order 5 in G / H . (d) Prove that G has an element y of order 5. (e) Let 1 6 = x H . Prove that the order of xy is neither 2 nor 5. (f) Prove that G is cyclic. (g) Prove that G = Z / 10 Z . (3 points) 2. Section 3.3, Exercise 8 on page 101.
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Unformatted text preview: (3 points) 3. Let G be a group, and suppose there is a homomorphism of G onto S 3 (the symmetric group of degree 3) with kernel K . Determine the number of subgroups of G which contain K , and show that exactly three of these subgroups are normal. (3 points) 4. Section 3.5, Exercise 2 on page 111. (2 points) 5. Section 4.1, Exercise 5(a) on page 116. (3 points) (5 problems, 14 points altogether)...
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