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  aHa1 = 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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 Spring '09
 DANBERWICK
 Algebra, Division, Integers, Natural number, Prime number, Euler, Cyclic group

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