# hw2 - 2 Section 4.1 Exercise 5(b on page 116(3 points 3...

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Math 5125 Friday, August 26 Second Homework Due 9:05 a.m., Friday September 2 1. Let G be a group with distinct normal subgroups A , B such that | G : A | = | G : B | = 2. (a) Prove that AB = G . (b) Prove that A / A B and B / A B are distinct normal subgroups of G / A B of order 2 and index 2. (c) Deduce that G has at least 3 subgroups of index 2. (You may assume that a group of order 4 is isomorphic to either Z / 4 Z or Z / 2 Z × Z / 2 Z .) (4 points)
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Unformatted text preview: 2. Section 4.1, Exercise 5(b) on page 116. (3 points) 3. Section 4.2, Exercise 5(a) on page 121. (3 points) 4. Let G be a group of order 75 acting on a set A of order 17. Suppose G ﬁxes no element of A (i.e. there is no orbit of order 1). Prove that G has a subgroup of order 15. (3 points) (4 problems, 13 points altogether)...
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