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Homework 08 - G = S 3 where S = x 1,x 2,x 3 is though of as...

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MATH 74 HOMEWORK 8 Due Wednesday April 17th at 3:10pm. Throughout, let G be a group. Many of the following problems are from (or adapted from) Herestein’s Topics in Algebra . (1) Let φ ( n ) be the Euler φ function, so φ ( n ) is the number of positive integers less than or equal to n that are relatively prime 1 to n . (a) Let ( Z /n Z ) × be the set of integers less than or equal to n relatively prime to n . Define a binary operation as multiplication modulo n . Prove that this defines a group. (b) Use part (a) and Lagrange’s theorem to show that for any positive integer n and any number a relatively prime to n , then a φ ( n ) 1 mod n . (2) Prove Fermat’s theorem, that if p is a prime number and a is any integer, then a p a mod p . (HINT: φ ( p ) = p - 1 and if a is less than p , a is relatively prime to p .) (3) Write out all the right and left cosets of H in G where (a) G is a cyclic group of order 10 with a generator a and H is the subgroup generated by a 2 . (b) G as in part (a) with H generated by a 5 . (c) G
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Unformatted text preview: G = S 3 where S = { x 1 ,x 2 ,x 3 } is though of as the group of bijective maps. H := { σ ∈ G | σ ( x 1 ) = x 1 } . (4) If H is a subgroup of G , then C ( H ) := { x ∈ G | xh = hx all h ∈ H } is called the centralizer of H . Prove that C ( H ) is a subgroup of G . (5) The center Z of a group G is defined as Z = { z ∈ G | xz = zx all x ∈ G } . Prove that Z is a subgroup of G . Can you recognize Z as C ( T ) for some subgroup T of G ? (6) If H is a subgroup of G , let N ( H ) = { a ∈ G | aHa-1 = H } . Prove that N ( H ) is a subgroup of G and that N ( H ) ⊇ H . (7) Prove that a subgroup of a cyclic group is a cyclic group. (8) How many generators does a cyclic group of order n have? (HINT: Remember Euler and his φ function.) 1 Integers x and y are relatively prime if the greatest common divisor of x and y is 1 1...
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