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I'm trying to relate the idea that in linear algebra any solution set $S$ of a given system of linear equation $Ax=b$ and $Ax_0=b$ then $S=\{x_0+k\mid k\in \ker A\}$, to some corresponding idea in group theory. So far I only know that if the kernel is used to divide the domain of a given homomorphism then the same image corresponding to the same coset of kernel. Please help if this is not a wrong question...

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If $f: G_1 \to G_2$ is a homomorphism of groups, then the set of solutions of $f(x)=b$ is $x_0K$, where $K=\ker f$ and $x_0$ is a particular solution.

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    $\begingroup$ Since only one operation is needed in my question so it's considered group, not ring right? $\endgroup$ Oct 26, 2018 at 10:55

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