question find the automorphism group of z16 express it as direct product of cyclic groups i do n

1. We want to find the automorphism group of Z16, which means we want to find all group homomorp

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Question: Find the automorphism group of Z16. Express it as a direct product of cyclic groups. I do not understand the answer. Please simplify it in a way that I would understand. How do I get the factorization of U(Z16) and similarly what is "phi([k]) = [tk]". Also, what is the composition of automorphisms corresponding to multiplication mod 16 in U(Z16)? How do I know U(Z16) is an abelian group? Then what is it with the direct computations? The part on the direct computations paragraph is very unfamiliar, and I am confused. Please explain. I do not understand. Answer: A map φ: Z16 → Z16 is a group homomorphism if and only if φ([k]) = [tk], where t ∈ {1, 3, 5, 7, 9, 11, 13, 15} = U(Z16). The composition of automorphisms corresponds to multiplication mod 16 in U(Z16). U(Z16) is an abelian group of order 8. So possible factorizations of U(Z16) as a product of cyclic groups are Z8, Z4 × Z2, or Z2 × Z2 × Z2. Direct computation shows that 1 = 9^2 = 7^2 and 1 = 3^4 = 11^4 = 5^4 = 13^4 in U(Z16). Since U(Z16) does not have elements of order 8 and U(Z16) has an element of order 4, we get U(Z16) ≅ Z4 × Z2.

          Question: Find the automorphism group of Z16. Express it as a direct product of cyclic groups. I do not understand the answer. Please simplify it in a way that I would understand. How do I get the factorization of U(Z16) and similarly what is "phi([k]) = [tk]". Also, what is the composition of automorphisms corresponding to multiplication mod 16 in U(Z16)? How do I know U(Z16) is an abelian group? Then what is it with the direct computations? The part on the direct computations paragraph is very unfamiliar, and I am confused. Please explain. I do not understand.

Answer: A map φ: Z16 → Z16 is a group homomorphism if and only if φ([k]) = [tk], where t ∈ {1, 3, 5, 7, 9, 11, 13, 15} = U(Z16). The composition of automorphisms corresponds to multiplication mod 16 in U(Z16). U(Z16) is an abelian group of order 8. So possible factorizations of U(Z16) as a product of cyclic groups are Z8, Z4 × Z2, or Z2 × Z2 × Z2.

Direct computation shows that 1 = 9^2 = 7^2 and 1 = 3^4 = 11^4 = 5^4 = 13^4 in U(Z16). Since U(Z16) does not have elements of order 8 and U(Z16) has an element of order 4, we get U(Z16) ≅ Z4 × Z2.

Added by Clara W.

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