Kernel Atomicity

Here, we show that the first isomorphism theorem, the orbit-stabilizer theorem, and the non-uniqueness of solutions of underdetermined linear systems are all manifestations of the same underlying algebraic property. We will call this algebraic property kernel atomicity. It arises principally because homomorphic maps induce partitions of their domain space into cosets, 'atoms' whose cardinalities are equal to that of the kernel.