Integral Cayley graphs and abelian rigidity in cores of cubelike graphs

A cubelike graph is a Cayley graph on an elementary abelian 2-group. The Cubelike Graph Core Conjecture of Mančinska, Pivotto, Roberson and Royle asserts that the core of every cubelike graph is cubelike. We prove the conjecture for cores that admit a normal abelian Cayley representation, that is, an abelian regular group of automorphisms that is normal in the full automorphism group. A retraction onto the core induces a commutative loop of exponent two whose translations are automorphisms. We show that its division maps are automorphisms as well, and that normality then forces the loop to be associative. We also determine the finite abelian groups G for which every integral Cayley graph on G is cubelike: they are the 2-groups with cyclic 2G, and the answer is the same when only connected cores are considered. On each remaining abelian 2-group, joins of copies of the Shrikhande graph give integral noncubelike cores that are covered by hypercubes but are not cores of cubelike graphs. We also give explicit binary coordinates under a square-zero biadditive condition and an orbit-equality criterion for a prescribed normal elementary abelian action with four orbits.