Efficient Isomorphism Testing for a Class of Group Extensions

TL;DR

This paper introduces an efficient algorithm for testing isomorphism in a specific class of nonabelian groups formed as extensions of abelian groups by cyclic groups, especially when their orders are coprime.

Contribution

The paper presents the first nearly linear time algorithm for isomorphism testing of certain nonabelian group extensions with coprime orders, expanding understanding beyond abelian groups.

Findings

Algorithm runs in almost linear time relative to group size.

Effective for groups where the abelian part's order is coprime with the cyclic extension.

Works in a black-box group setting.

Abstract

The group isomorphism problem asks whether two given groups are isomorphic or not. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. In this paper we study this problem for a class of groups corresponding to one of the simplest ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group A by a cyclic group of order m. We present an efficient algorithm solving the group isomorphism problem for all the groups of this class such that the order of A is coprime with m. More precisely, our algorithm runs in time almost linear in the orders of the input groups and works in the general setting where the groups are given as black-boxes.