Efficient Algorithms for Abelian Group Isomorphism and Related Problems

We consider the problem of determining if two finite groups are isomorphic. The groups are assumed to be represented by their multiplication tables. We present an algorithm that determines if two Abelian groups with n elements each are isomorphic. The running time of this algorithm is O(n log p), where p is the smallest prime not dividing n. When n is odd, this algorithm runs in linear time; in general, it takes time at most O(n log log n), improving upon an algorithm with O(n log n) running time due to Vikas [13]. Our Abelian group isomorphism algorithm is a byproduct of an algorithm that computes the orders of all elements in any group (not necessarily Abelian) of size n in time O(n log p), where p is the smallest prime not dividing n. We also give an O(n) algorithm for determining if a group of size n, described by its multiplication table, is Abelian.

[1]  N. F. Kuzennyi.  Isomorphism of semidirect products , 1974 .

[2]  R. Lipton,et al.  The Complexity of Word and Isomorphism Problems for Finite Groups. , 1977 .

[3]  Ravi Kannan,et al.  Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix , 1979, SIAM J. Comput..

[4]  John H. Hodges.  Scalar polynomial equations for matrices over a finite field , 1958 .

[5]  Endre Szemerédi,et al.  On the Complexity of Matrix Group Problems I , 1984, FOCS.

[6]  Gary L. Miller.  Graph Isomorphism, General Remarks , 1979, J. Comput. Syst. Sci..

[7]  I. Herstein,et al.  Topics in algebra , 1964 .

[8]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[9]  Narayan Vikas.  An O(n) Algorithm for Abelian p-Group Isomorphism and an O(n log n) Algorithm for Abelian Group Isomorphism , 1996, J. Comput. Syst. Sci..

[10]  Yechezkel Zalcstein,et al.  On Isomorphism Testing of a Class of 2-Nilpotent Groups , 1991, J. Comput. Syst. Sci..

[11]  László Babai,et al.  Local expansion of vertex-transitive graphs and random generation in finite groups , 1991, STOC '91.

[12]  Victor Shoup,et al.  A computational introduction to number theory and algebra , 2005 .

[13]  Allan K. Steel,et al.  Algorithm for the Computation of Canonical Forms of Matrices over Fields , 1997, J. Symb. Comput..

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

[15]  J. Köbler,et al.  The Graph Isomorphism Problem: Its Structural Complexity , 1993 .

[16]  T. Apostol.  Introduction to analytic number theory , 1976 .

[17]  D. Shanks.  Class number, a theory of factorization, and genera , 1971 .

[18]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[19]  Harry G. Mairson.  Some new upper bounds on the generation of prime numbers , 1977, CACM.

[20]  J. Pomfret.  Similarity of matrices over finite rings , 1973 .

[21]  Michael J. Smith.  Computing automorphisms of finite soluble groups , 1996, Bulletin of the Australian Mathematical Society.

[22]  Johannes A. Buchmann,et al.  Computing the structure of a finite abelian group , 2005, Math. Comput..

[23]  E. A. O'Brien.  Isomorphism Testing for p-Groups , 1994, J. Symb. Comput..

[24]  Helmut Hasse.  The Class Number , 1980 .

[25]  George E. Collins,et al.  Algorithms for the Solution of Systems of Linear Diophantine Equations , 1982, SIAM J. Comput..

[26]  Jacobo Torán,et al.  Solvable Group Isomorphism , 2004, Computational Complexity Conference.