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# hw3 - two in S 6(2 points 3 Let G be a group of order 21...

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Math 5125 Friday, September 2 Third Homework Due 9:05 a.m., Friday September 9 1. Let n Z + , let G be a group with a subgroup H of index n , and let g G be an element of order 2. Suppose 4 n , and that whenever θ : G S n is a homomorphism, then θ ( G ) A n . Prove that gxH = xH for some x G . (Consider the action of G on the left cosets of H and in particular the cycle decomposition of g with respect to this action.) (3 points) 2. Determine the size of the conjugacy class and centralizer for each element of order
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Unformatted text preview: two in S 6 . (2 points) 3. Let G be a group of order 21 with Z ( G ) = 1. Prove that G has exactly 5 conjugacy classes. (3 points) 4. Let G be a group with exactly two elements x , y of order 2. Prove that either x ∈ Z ( G ) or | G : C G ( x ) | = 2. Deduce that C G ( x ) ± G . (3 points) 5. Section 4.4, Exercise 3 on page 137. (2 points) (5 problems, 13 points altogether)...
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