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›› 2012, Vol. 27 ›› Issue (4): 687-701.

• Special Section on Theoretical Computer Science •

### On Isomorphism Testing of Groups with Normal Hall Subgroups

You-Ming Qiao1 (乔友明), Jayalal Sarma M.N.2, and Bang-Sheng Tang1 (唐邦晟)

1. 1. Institute for Theoretical Computer Science, Tsinghua University, Beijing 100084, China;
2. Department of Computer Science and Engineering, Indian Institute of Technology Madras, Chennai 600036, India
• Received:2011-03-13 Revised:2012-02-29 Online:2012-07-05 Published:2012-07-05
• Supported by:

The work was supported in part by the National Natural Science Foundation of China under Grant No. 60553001, and the National Basic Research 973 Program of China under Grant Nos. 2007CB807900 and 2007CB807901.

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 Zpd, p a prime, determine if there exists an automorphism Φ:HH, such that the induced representation ρΦ=ρ ο Φ and τ are equivalent, in time poly(|H|, pd).

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