Compatible filters with isomorphism testing

Abstract

Like the lower central series of a nilpotent group, filters generalize the connection between nilpotent groups and graded Lie rings. However, unlike the case with the lower central series, the associated graded Lie ring may share few features with the original group: e.g. the associated Lie ring may be trivial or arbitrarily large. We determine properties of filters such that the Lie ring and group are in bijection. We prove that, under such conditions, every isomorphism between groups is induced by an isomorphism between graded Lie rings.

Incorporating Weisfeiler-Leman into algorithms for group isomorphism

“…We introduce a colored (hyper-)graph based on an algebraic data structure known as a recursivelyrefineable filter which identifies abelian groups and vector spaces layered together to form the structure of a finite group. Filters have been useful in several isomorphism tests [W3,M3,M1,BOW]. As the name suggests, filters can be refined, and with each refinement the cost T (n) of isomorphism testing decreases after refinement to a function in O(T (n) 1/c ), for some c 2.…”