More characteristic subgroups, Lie rings, and isomorphism tests for p-groups

Abstract

Abstract. We introduce three families of characteristic subgroups that refine the traditional verbal subgroup filters, such as the lower central series, to an arbitrary length. We prove that a positive logarithmic proportion of finite p-groups admit at least five such proper nontrivial characteristic subgroups whereas verbal and marginal methods explain only one. The placement of these subgroups in the lattice of subgroups is naturally recorded by a filter over an arbitrary commutative monoid M and induces an M-graded Lie ring. These Lie rings permit an efficient specialization of the nilpotent quotient algorithm to construct automorphisms and decide isomorphism of finite p-groups. As a demonstration, we identify some large families of p-groups that are worst-case examples for the traditional nilpotent quotient algorithm but run in polynomial time when using our new filters.