Abstract Algebra I · Homomorphisms, kernels, and normal subgroups
A cyclic homomorphism from one formula
Problem
Fix an integer $a$ and a positive integer $n$. Define $\varphi:\mathbb Z\to\mathbb Z/n\mathbb Z$ by $\varphi(k)=[ak]_n$. Prove that $\varphi$ is a homomorphism and show that $\ker\varphi=\frac{n}{\gcd(a,n)}\mathbb Z$ and $|\operatorname{im}\varphi|=\frac{n}{\gcd(a,n)}$.
Hint
Check $[a(k+\ell)]_n=[ak]_n+[a\ell]_n$.
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