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(Solved): 0. (i) Prove that \( \phi:(\mathbb{Z},+) \rightarrow Z_{n}=\langle x\rangle \) with \( \phi(m)=x^{m ...




0. (i) Prove that \( \phi:(\mathbb{Z},+) \rightarrow Z_{n}=\langle x\rangle \) with \( \phi(m)=x^{m} \) is a homomorphism;
(i
0. (i) Prove that \( \phi:(\mathbb{Z},+) \rightarrow Z_{n}=\langle x\rangle \) with \( \phi(m)=x^{m} \) is a homomorphism; (ii) Find the kernel \( K=\phi^{-1}\left(1_{Z_{n}}\right) \) and the image \( \phi(\mathbb{Z}) \); (iii) Use the results from (i) and (ii) to conclude that \( \mathbb{Z} /(n \mathbb{Z}) \cong Z_{n} \).


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given solution - Answer 10(i) given that ? : (Z, +) ? Zn with ? (m) = xm where Zn = (x) let m, n ? Z then ? (m + n) = x(m+n) = xmxn = ?(m). ?(n) so is
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