Finding central decompositions of p-groups

TL;DR

This paper presents polynomial-time algorithms, including probabilistic and deterministic methods, for finding maximum size central decompositions of finite p-groups and nilpotent Lie rings of class 2, introducing new invariants.

Contribution

It introduces efficient algorithms and new invariants for decomposing p-groups and Lie rings, advancing computational group theory.

Findings

Algorithms run in polynomial time for p-groups of class 2

Probabilistic routines are effective and can be replaced by deterministic algorithms for small p

New group isomorphism invariants and characteristic subgroups are introduced

Abstract

Polynomial-time algorithms are given to find a central decomposition of maximum size for a finite p-group of class 2 and for a nilpotent Lie ring of class 2. The algorithms use Las Vegas probabilistic routines to compute the structure of finite *-rings and also the Las Vegas C-MeatAxe. When p is small, the probabilistic methods can be replaced by deterministic polynomial-time algorithms. The methods introduce new group isomorphism invariants including new characteristic subgroups.