# On Isomorphism Testing of Groups with Normal Hall Subgroups

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## Abstract

A normal Hall subgroup *N* of a group *G* is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives *H* which also forms a subgroup of *G*. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall (STACS 2009), we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem, which asks whether two linear subspaces are the same up to permutation of coordinates. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai *et al*. (SODA 2011). Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations ρ and τ of a group *H* over \( \mathbb{Z}_p^d \), *p* a prime, determine if there exists an automorphism : *H → H*, such that the induced representation ρ𝜙 = ρ *◦* 𝜙 and τ are equivalent, in time poly(*|H|, p* ^{ d }).

## Keywords

group isomorphism normal Hall subgroup isomorphism testing problem complexity## Preview

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## References

- [1]Dehn M. Über unendliche diskontinuierliche gruppen.
*Mathematische Annalen*, 1911, 71: 116–144.MathSciNetGoogle Scholar - [2]Adian S. The unsolvability of certain algorithmic problems in the theory of groups.
*Trudy Moskov. Math. Obshch*, 1957, 6: 231–298.Google Scholar - [3]Babai L, Szemerédi E. On the complexity of matrix group problems
*i*. In*Proc. IEEE Annual Symposium on Foundations of Computer Science*(*FOCS*), West Palm Beach, Florida, USA, October 24–26, 1984, pp.229–240.Google Scholar - [4]Köbler J, Schöning U, Torán J. The Graph Isomorphism Problem: Its Structural Complexity. Boston: Birkhauser, 1993.zbMATHGoogle Scholar
- [5]Chattopadhyay A, Torán J, Wagner F. Graph isomorphism is not
*AC*^{0}reducible to group isomorphism. In*Proc. Annual Conference on Foundations of Software Technology and Theoretical Computer Science*(*FSTTCS 2010*), Chennai, India, Dec. 15–18, 2010, pp.317–326.Google Scholar - [6]Babai L, Luks E M. Canonical labeling of graphs. In
*Proc. the 15th Annual ACM Symposium on Theory of Computing*(*STOC*), Boston, Massachusetts, USA, April 25–27, 1983, pp.171–183.Google Scholar - [7]Miller G L. On the
*n*log*n*isomorphism technique. In*Proc. the 10th Annual ACM Symposium on Theory of Computing*, San Diego, California, USA, May 1–3, 1978, pp.51–58.Google Scholar - [8]Lipton R J, Snyder L, Zalcstein Y. The complexity of word and isomorphism problems for finite groups. Technical Report, John Hopkins, 1976.Google Scholar
- [9]Savage C. An
*O*(*n*^{2}) algorithm for abelian group isomorphism. Technical Report, North Carolina State University, 1980.Google Scholar - [10]Vikas N. An
*O*(*n*) algorithm for abelian*p*-group isomorphism and an*O*(*n*log*n*) algorithm for abelian group isomorphism.*Journal of Computer and System Sciences*, 1996, 53(1): 1–9.MathSciNetzbMATHCrossRefGoogle Scholar - [11]Kavitha T. Linear time algorithms for abelian group isomorphism and related problems.
*Journal of Computer and System Sciences*, 2007, 73(6): 986–996.MathSciNetzbMATHCrossRefGoogle Scholar - [12]Gall F L. Efficient isomorphism testing for a class of group extensions. In
*Proc. the 26th International Symposium on Theoretical Aspects of Computer Science*(*STACS 2009*), Freiburg, Germany, February 26–28, 2009, pp.625–636.Google Scholar - [13]Wilson J B. Decomposing
*p*-groups via Jordan algebras.*Journal of Algebra*, 2009, 322(8): 2642–2679.MathSciNetzbMATHCrossRefGoogle Scholar - [14]Wilson J B. Finding central decompositions of
*p*-groups.*Journal of Group Theory*, 2009, 12(6): 813–830.MathSciNetzbMATHCrossRefGoogle Scholar - [15]Kayal N, Nezhmetdinov T. Factoring groups efficiently. In
*Proc. the 36th International Colloquium on Automata, Languages and Programming*(*ICALP 2009*), Rhodes, Greece, July 5–12, 2009, pp.585–596.Google Scholar - [16]Wilson J B. Finding direct product decompositions in polynomial time. 2010. http://arxiv.org/pdf/1005.0548.pdf.
- [17]Taunt D R. Remarks on the isomorphism problem in theories of construction of finite groups.
*Mathematical Proceedings of the Cambridge Philosophical Society*, 1955, 51(1): 16–24.MathSciNetzbMATHCrossRefGoogle Scholar - [18]Menegazzo F. The number of generators of a finite group.
*Irish Math. Soc. Bulletin*, 2003, 50: 117–128.MathSciNetzbMATHGoogle Scholar - [19]Babai L. Equivalence of linear codes. Technical Report, University of Chicago, 2010.Google Scholar
- [20]Babai L, Codenotti P, Grochow J, Qiao Y. Towards efficient algorithm for semisimple group isomorphism. In
*Proc. ACMSIAM Annual Symposium of Discrete Algorithms*(*SODA*), San Francisco, California, USA, January 23–25, 2011.Google Scholar - [21]Holt D F, Eick B, O’Brien E A. Handbook of Computational Group Theory. London: Chapman and Hall/CRC, 2005.zbMATHCrossRefGoogle Scholar
- [22]Rotman J J. An Introduction to the Theory of Groups (4th edition). Springer-Verlag, 1995.Google Scholar
- [23]Serre J P. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977.zbMATHCrossRefGoogle Scholar
- [24]Rónyai L. Computing the structure of finite algebras.
*Journal of Symbolic Computation*, 1990, 9(3): 355–373.MathSciNetzbMATHCrossRefGoogle Scholar - [25]Shoup V. On the deterministic complexity of factoring polynomials over finite fields.
*Information Processing Letters*, 1990, 33(5): 261–267.MathSciNetzbMATHCrossRefGoogle Scholar - [26]Steel A. A new algorithm for the computation of canonical forms of matrices over fields.
*Journal of Symbolic Computation*, 1997, 24(3–4): 409–432.MathSciNetzbMATHCrossRefGoogle Scholar - [27]Seress A. Permutation Group Algorithms. Cambridge: Cambridge University Press, 2003.zbMATHCrossRefGoogle Scholar
- [28]Babai L. Coset intersection in moderately exponential time.
*Chicago Journal of Theoretical Computer Science*, to appear.Google Scholar - [29]Luks E M. Hypergraph isomorphism and structural equivalence of Boolean functions. In
*Proc. the 31st Annual ACM Symposium on Theory of Computing*, Atlanta, Georgia, USA, May 1–4, 1999, pp.652–658.Google Scholar - [30]Kantor W M, Luks E M, Mark P D. Sylow subgroups in parallel.
*Journal of Algorithms*, 1999, 31(1): 132–195.MathSciNetzbMATHCrossRefGoogle Scholar - [31]Babai L, Qiao Y. Polynomial-time isomorphism test for groups with abelian Sylow towers. In
*Proc. the 29th International Symposium on Theoretical Aspects of Computer Science*, Pairs, France, Feb. 28-March 3, 2012, pp.453–464.Google Scholar - [32]Petrank E, Roth R M. Is code equivalence easy to decide?
*IEEE Trans. Information Theory*, 1997, 43(5): 1602–1604.MathSciNetzbMATHCrossRefGoogle Scholar - [33]Buchmann J, Schmidt A. Computing the structure of a finite abelian group.
*Mathematics of Computation*, 2005, 74(252): 2017–2026.MathSciNetzbMATHCrossRefGoogle Scholar - [34]Ranum A. The group of classes of congruent matrices with application to the group of isomorphisms of any abelian group.
*Transactions of the American Mathematical Society*, 1907, 8(1): 71–91.MathSciNetzbMATHCrossRefGoogle Scholar