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.

Literature Survey on Non-Associative Rings and Developments

“…In 2011, Khukhro, Makarenko and Shumyatsky [45] developed a Lie ring theory which is used for studying groups G and Lie rings L with a metacyclic Frobenius group of automorphisms F H. Wilson [258] in 2013 introduced three families of characteristic subgroups that refined the traditional verbal subgroup filters, such as the lower central series, to an arbitrary length. It was proved that a positive logarithmic proportion of finite p-groups admit at least five such proper nontrivial characteristic subgroups whereas verbal and marginal methods explained only one.…”

Efficient characteristic refinements for finite groups

“…Filters and prefilters. The formulas found in [Wil13] are for commutative monoids, but for computational feasibility, we concentrate only on N d with the lexicographical order. We note that every filter φ : N d → 2 G induces a boundary filter ∂φ : N d → 2 G , where ∂φ s = φ s+t | 0 ≺ t .…”

“…In [Wil13,Wil15], sources of new subgroups were suggested. We use these in our testing, but we remark that our methods can use any source of refinements, and hence, any prefilter.…”

“…It is in these nonassociative rings we begin to find more characteristic structure in G [Wil13,Wil15]. For example, the Jacobson radical acts on the homogeneous components and yields characteristic subgroups (for Der(•), this is done in the associative enveloping algebra).…”

“…New sources for computable characteristic subgroups of p-groups were uncovered in [Wil13,Wil15]. In addition, it was shown that the inclusion of new characteristic subgroups induced more subgroups and gave formulas to automate this process of refining.…”