# Low-complexity computations for nilpotent subgroup problems

## TL;DR

This paper develops efficient algorithms for key subgroup problems in nilpotent groups, achieving low-complexity solutions using TC0 circuits and logspace, even with compressed inputs.

## Contribution

It introduces novel low-complexity algorithms for subgroup problems in nilpotent groups, including handling compressed inputs, with uniform solutions in circuit and logspace models.

## Key findings

- Algorithms run in TC0 circuits and logspace for nilpotent groups.
- Handling compressed inputs with quartic time algorithms.
- Provides efficient solutions for subgroup conjugacy, normalizer, isolator, coset intersection, and torsion subgroup.

## Abstract

We solve the following algorithmic problems using TC0 circuits, or in logspace and quasilinear time, uniformly in the class of nilpotent groups with bounded nilpotency class and rank: subgroup conjugacy, computing the normalizer and isolator of a subgroup, coset intersection, and computing the torsion subgroup. Additionally, if any input words are provided in compressed form as straight-line programs or in Mal'cev coordinates the algorithms run in quartic time.