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Beating the Generator-Enumeration Bound for Solvable-Group Isomorphism

Published:03 May 2020Publication History
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Abstract

We consider the isomorphism problem for groups specified by their multiplication tables. Until recently, the best published bound for the worst-case was achieved by the nlogp n + O(1) generator-enumeration algorithm where n is the order of the group and p is the smallest prime divisor of n. In previous work with Fabian Wagner, we showed an n(1 / 2) logp n + O(log n / log log n) -time algorithm for testing isomorphism of p-groups by building graphs with degree bounded by p + O(1) that represent composition series for the groups and applying Luks’ algorithm for testing isomorphism of bounded-degree graphs.

In this work, we extend this improvement to the more general class of solvable groups to obtain an n(1 / 2) logp n + O(log n / log log n) -time algorithm. In the case of solvable groups, the composition factors can be large which prevents previous methods from outperforming the generator-enumeration algorithm. Using Hall’s theory of Sylow bases, we define a new object that generalizes the notion of a composition series with small factors but exists even when the composition factors are large. By constructing graphs that represent these objects and running Luks’ algorithm, we obtain our algorithm for solvable-group isomorphism. We also extend our algorithm to compute canonical forms of solvable groups while retaining the same complexity.

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 12, Issue 2
        June 2020
        138 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3382781
        Issue’s Table of Contents

        Copyright © 2020 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 3 May 2020
        • Revised: 1 February 2020
        • Accepted: 1 February 2020
        • Received: 1 January 2014
        Published in toct Volume 12, Issue 2

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