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Groups Complexity Cryptology

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Isomorphism in expanding families of indistinguishable groups

Mark L. Lewis
  • Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
  • :
/ James B. Wilson
  • Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
  • :
Published Online: 2012-05-01 | DOI: https://doi.org/10.1515/gcc-2012-0008


For every odd prime and every integer , there is a Heisenberg group of order that has pairwise nonisomorphic quotients of order . Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most . They are also directly and centrally indecomposable and of the same indecomposability type. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.

Keywords: -group; group isomorphism; polynomial-time; asymptotic group theory

Received: 2010-10-27

Published Online: 2012-05-01

Published in Print: 2012-05-01

Citation Information: Groups - Complexity - Cryptology. Volume 4, Issue 1, Pages 73–110, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2012-0008, May 2012

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