# The term to describe linear algebra's $\textrm{general solution }=\textrm{particular sol. + homogeneous sol.}$ in group theory

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...

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.