Abstract
A normal Hall subgroup $N$ of a group $G$ is a normal subgroup with its order coprime with its index. SchurZassenhaus 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 SchurZassenhaus 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. 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. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations rho and tau of a group $H$ over Z_p^d , p a prime, determine if there exists an automorphism phi:H > H, such that the induced representation rho_phi=rho o phi and tau are equivalent, in time poly(H,p^d).
BibTeX  Entry
@InProceedings{qiao_et_al:LIPIcs:2011:3044,
author = {Youming Qiao and Jayalal Sarma M.N. and Bangsheng Tang},
title = {{On Isomorphism Testing of Groups with Normal Hall Subgroups}},
booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
pages = {567578},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897255},
ISSN = {18688969},
year = {2011},
volume = {9},
editor = {Thomas Schwentick and Christoph D{\"u}rr},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2011/3044},
URN = {urn:nbn:de:0030drops30443},
doi = {10.4230/LIPIcs.STACS.2011.567},
annote = {Keywords: Group Isomorphism Problem, Normal Hall Subgroups, Computational Complexity}
}
Keywords: 

Group Isomorphism Problem, Normal Hall Subgroups, Computational Complexity 
Collection: 

28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) 
Issue Date: 

2011 
Date of publication: 

11.03.2011 