On the Group and Color Isomorphism Problems

TL;DR

This paper explores the complexity relationship between group and color isomorphism problems, identifying the projective special linear group as a key obstacle to developing faster algorithms for group isomorphism.

Contribution

It establishes reductions between group and color isomorphism problems excluding alternating groups, highlighting the projective special linear group as the main computational barrier.

Findings

Group isomorphism reduces to color isomorphism with non-alternating composition factors.

Color isomorphism with certain factors reduces back to a generalized group isomorphism.

Projective special linear group is identified as the primary obstacle to faster algorithms.

Abstract

In this paper, we prove results on the relationship between the complexity of the group and color isomorphism problems. The difficulty of color isomorphism problems is known to be closely linked to the the composition factors of the permutation group involved. Previous works are primarily concerned with applying color isomorphism to bou nded degree graph isomorphism, and have therefore focused on the alternating composit ion factors, since those are the bottleneck in the case of graph isomorphism. We consider the color isomorphism problem with composition factors restricted to those other than the alternating group, show that group isomorphism reduces in n^(O(log log n)) time to this problem, and, conversely, that a special case of this color isomorphism problem reduces to a slight generalization of group isomorphism. We then sharpen our results by identifying the projective special linear group as the main obstacle to faster algorithms for group isomorphism and prove that the aforementioned reduc tion from group isomorphism to color isomorphism in fact produces only cyclic and projective special linear factors. Our results demonstrate that, just as the alternatin g group was a barrier to faster algorithms for graph isomorphism for three decades, the projective special linear group is an obstacle to faster algorithms for group isomorphism.