Kernel Atomicity

TL;DR

This paper introduces the concept of kernel atomicity, unifying several fundamental algebraic theorems as manifestations of a common underlying property related to kernels and cosets.

Contribution

It defines kernel atomicity as an underlying algebraic property that explains the first isomorphism theorem, orbit-stabilizer theorem, and solution non-uniqueness in linear systems.

Findings

Unifies key algebraic theorems under kernel atomicity.

Shows how homomorphic maps induce partitions into cosets.

Highlights the role of kernels in algebraic structures.

Abstract

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.