Constructive membership in black-box groups

Abstract We present an algorithm to reduce the constructive membership problem for a black-box group G to three instances of the same problem for involution centralizers in G. If G is a finite simple group of Lie type in odd characteristic, then this reduction can be performed in (Monte Carlo) polynomial time.

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

[2]  Igor Pak,et al.  The product replacement algorithm is polynomial , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[3]  Christopher Parker,et al.  Recognising simplicity of black-box groups by constructing involutions and their centralisers , 2010 .

[4]  D. Gorenstein,et al.  The Classification of the Finite Simple Groups , 1983 .

[5]  M. Aschbacher.  Finite Group Theory: Representations of groups on groups , 2000 .

[6]  László Babai,et al.  Black-box recognition of finite simple groups of Lie type by statistics of element orders , 2002 .

[7]  E. A. O'Brien,et al.  Constructive recognition of $\mathrm{PSL}(2, q)$ , 2005 .

[8]  A. Borovik.  Probabilistic Recognition of Orthogonal and Symplectic Groups : Corrections , 2001 .

[9]  William M. Kantor,et al.  On constructive recognition of a black box PSL , 2001 .

[10]  William M. Kantor,et al.  On the Probability That a Group Element is p-Singular , 1995 .

[11]  Robert A. Wilson,et al.  Recognising simplicity of black-box groups , 2005 .

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

[13]  John N. Bray.  An improved method for generating the centralizer of an involution , 2000 .

[14]  C. Praeger,et al.  Generalised Sifting in Black-Box Groups , 2005, math/0501346.

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

[16]  A. Borovik.  Centralisers of Involutions in Black Box Groups , 2001, math/0110233.

[17]  Cheryl E. Praeger,et al.  A black-box group algorithm for recognizing finite symmetric and alternating groups, I , 2003 .

[18]  L. E. Dickson.  Group Theory: 5 , 1904 .

[19]  C. R. Leedham-Green,et al.  Constructive recognition of classical groups in odd characteristic , 2007 .

[20]  C. Sims.  Computational methods in the study of permutation groups , 1970 .

[21]  William M. Kantor,et al.  Black Box Classical Groups , 2001 .

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

[23]  Martin W. Liebeck,et al.  Finding the characteristic of a group of Lie type , 2007 .

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

[25]  Scott H. Murray,et al.  Generating random elements of a finite group , 1995 .

[26]  William M. Kantor,et al.  Fast constructive recognition of black box orthogonal groups , 2006 .

[27]  Charles C. Sims,et al.  Computation with finitely presented groups , 1994, Encyclopedia of mathematics and its applications.

[28]  D. Gorenstein,et al.  The Classification of the Finite Simple Groups, Number 2 , 1995 .

[29]  Cheryl E. Praeger,et al.  A Recognition Algorithm For Classical Groups Over Finite Fields , 1998 .

[30]  Charles R. Leedham-Green,et al.  Calculating the order of an invertible matrix , 1995, Groups and Computation.

[31]  László Babai,et al.  Fast Monte Carlo Algorithms for Permutation Groups , 1995, J. Comput. Syst. Sci..

[32]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[33]  Aner Shalev,et al.  The probability of generating a finite simple group , 1995 .

[34]  Steve Linton.  The Art and Science of Computing in Large Groups , 1995 .