Polynomial-Time Isomorphism Test for Groups with No Abelian Normal Subgroups - (Extended Abstract)

We consider the problem of testing isomorphism of groups of order n given by Cayley tables. The trivial nlogn bound on the time complexity for the general case has not been improved upon over the past four decades. We demonstrate that the obstacle to efficient algorithms is the presence of abelian normal subgroups; we show this by giving a polynomial-time isomorphism test for groups without nontrivial abelian normal subgroups. This concludes a project started by the authors and J. A. Grochow (SODA 2011). Two key new ingredient are: (a) an algorithm to test permutational isomorphism of permutation groups in time, polynomial in the order and simply exponential in the degree; (b) the introduction of the "twisted code equivalence problem," a generalization of the classical code equivalence problem by admitting a group action on the alphabet. Both of these problems are of independent interest.

[1]  John E. Hopcroft,et al.  Polynomial-time algorithms for permutation groups , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[2]  François Le Gall,et al.  Efficient Isomorphism Testing for a Class of Group Extensions , 2008, STACS.

[3]  Robert Steinberg,et al.  Generators for Simple Groups , 1962, Canadian Journal of Mathematics.

[4]  Michio Suzuki.  Group Theory II , 2012 .

[5]  J. Neubüser,et al.  On a programme for the determination of the automorphism group of a finite group , 1970 .

[6]  Michael Aschbacher,et al.  Some applications of the first cohomology group , 1984 .

[7]  Á. Seress.  Permutation Group Algorithms , 2003 .

[8]  Joshua A. Grochow,et al.  Code equivalence and group isomorphism , 2011, SODA '11.

[9]  L. Babai,et al.  Groups St Andrews 1997 in Bath, I: A polynomial-time theory of black box groups I , 1999 .

[10]  Ákos Seress,et al.  Computing the Fitting subgroup and solvable radical for small-base permutation groups in nearly linear time , 1995, Groups and Computation.

[11]  László Babai,et al.  Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers , 2012, STACS.

[12]  Eugene M. Luks,et al.  Hypergraph isomorphism and structural equivalence of Boolean functions , 1999, STOC '99.

[13]  P. Cameron.  FINITE PERMUTATION GROUPS AND FINITE SIMPLE GROUPS , 1981 .

[14]  James B. Wilson.  Decomposing p-groups via Jordan algebras , 2007, 0711.0201.

[15]  Eugene M. Luks,et al.  Polynomial-time normalizers for permutation groups with restricted composition factors , 2002, ISSAC '02.

[16]  Derek F. Holt,et al.  Automorphism group computation and isomorphism testing in finite groups , 2003, J. Symb. Comput..

[17]  Donald E. Knuth.  Efficient representation of perm groups , 1991, Comb..

[18]  Charles C. Sims,et al.  Computation with permutation groups , 1971, SYMSAC '71.

[19]  Jayalal Sarma,et al.  On Isomorphism Testing of Groups with Normal Hall Subgroups , 2011, STACS.

[20]  Telikepalli Kavitha.  Linear time algorithms for Abelian group isomorphism and related problems , 2007, J. Comput. Syst. Sci..

[21]  James B. Wilson.  Finding central decompositions of p-groups , 2008, 0801.3434.

[22]  D. Robinson.  A Course in the Theory of Groups , 1982 .

[23]  Gary L. Miller,et al.  On the nlog n isomorphism technique (A Preliminary Report) , 1978, STOC.