Journal of Siberian Federal University. Mathematics & Physics, 7(), 8 9 УДК 7.9 Hall s Polynomials of Finite Two -Generator Groups of Exponent Seven Alexander A. Kuznetsov Konstantin V. Safonov Institute of Computer Science and Telecommunications, Siberian State Aerospace University, Krasnoyarsky Rabochy,, Krasnoyarsk, Russia Received.., received in revised form 8.., accepted.. Let B k = B (, 7, k) be the largest two-generator finite group of exponent 7 and nilpotency class k. Hall s polynomials of B k for k are calculated. Keywords: periodic group, collection process, Hall s polynomials. Let B k = B (,7,k) be the largest two-generator finite group of exponent 7 and nilpotency class k. In this class, the largest group is the group B 8, which has the order 7 []. For each B k a power commutator presentation is obtained []. Let a x...axn n and a y...ayn n be two arbitrary elements in the group B k recorded in the commutator form. Then their product is equal a x...axn n a y...ayn n = a z...azn n. Powers z i are to be found based on the collection process (see [,]) which is implemented in the computer algebra systems GAP and MAGMA. Furthermore, there is an alternative method for calculating products of elements of the group, proposed by Hall (see []). Hall showed that z i are polynomial functions (over the field Z 7 in this case), depending on the variables x,...,x i,y,...,y i, which is now called Hall s polynomials. According to [] z i = x i + y i + p i (x,...,x i,y,...,y i ). Hall s polynomials are necessary in solving problems that require multiple products of the elements of the group. Study of the structure of the Cayley graph of some group is one of these problems [, ]. The computational experiments carried out on the computer in two-generator groups of exponent five (see [7]) showed that the method of Hall s polynomials has an advantage over the traditional collection process. Therefore, there is a reason to believe that the use of polynomials would be preferable than the collection process in the study of Cayley graphs of B k groups. It should also be noted that this method is easily software-implemented including multiprocessor computer systems. Previously unknown Hall s polynomials of B k are calculated within the framework of this paper. For k > polynomials are calculated similarly but their output takes considerably more space so it makes impossible to verify the proof without use of computers. The main result of this paper is Theorem. Let a x...axn n and a y...ayn n be two arbitrary elements of the group B k recorded in the commutator form where k N и k. Then their product is equal a x...axn n a y...ayn n = alex_kuznetsov8@mail.ru safonovkv@rambler.ru c Siberian Federal University. All rights reserved 8
Alexander A.Kuznetsov, Konstantin V.Safonov Hall s Polynomials of Finite Two -Generator Groups... a z...azn n, where z i Z 7, are Hall s polynomials given by formulas ( ) for k =, ( ) for k =, ( ) for k = and ( 8) for k =. z = x + y, () z = x + y, () z = x + y + x y, () z = x + y + x y + x y + x y, () z = x + y + x y + x y + x y + x y y, () z = x + y + x y + x y + x y + x y + x y + x y, () z 7 = x 7 + y 7 + x y + x y + x y + x y + x y + x y + x y y + x y y + x y y, (7) z 8 = x 8 + y 8 + x y + x y + x y + x y + x y + x y + x y y + x y y + x y y. (8). Proof of the Theorem Let s calculate the Hall s polynomials for the group B. Dealing with this group we also obtain polynomials for groups B, B and B so no need to study separately these cases. When k < commutators which has a weight is more than k are not considered because of they are definitionally equal to the group identity. Using GAP we obtain a power commutator presentation of B. Commutators of weight : a, a generators of the group. Commutators of weight : a = [a,a ]. Commutators of weight : Commutators of weight : a = [a,a ] = [a,a,a ], a = [a,a ] = [a,a,a ]. a = [a,a ] = [a,a,a,a ], a 7 = [a,a ] = [a,a,a,a ], a 8 = [a,a ] = [a,a,a,a ]. List of defining relations R for commutators: a 7 i = ( i 8), [a,a ] = a, [a,a ] = a, [a,a ] = a, [a,a ] = a, [a,a ] = a 7, [a,a ] =, [a,a ] = a 7, [a,a ] = a 8, [a,a ] =, [a,a ] =, [a,a ] =, [a,a ] =, [a,a ] =, [a,a ] =, [a,a ] =, [a 7,a ] =, [a 7,a ] =, [a 7,a ] =, [a 7,a ] =, [a 7,a ] =, [a 7,a ] =, [a 8,a ] =, [a 8,a ] =, [a 8,a ] =, [a 8,a ] =, [a 8,a ] =, [a 8,a ] =, [a 8,a 7 ] =. 87
Alexander A.Kuznetsov, Konstantin V.Safonov Hall s Polynomials of Finite Two-Generator Groups... Thus, B = a,a,a,a,a,a,a 7,a 8 R. Each element of the group is expressed uniquely as a normal commutator word: g B g = a x ax ax ax ax ax ax7 7 ax8 8, x i Z 7. Sometimes we will write g = (x,...,x 8 ). In order to determine the functions z i first we need to calculate the products of a y j ax i for all i < j 8, x,y =,,,,,. For the pair (j,i) it is required to find the interpolation polynomial for each of the 8 commutators by the values of the product (y,x). Let s start with the first pair a y ax : a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,), a a = (,,,,,,,). Let s write: a y ax = a x a y af(,) (x,y) a f(,) (x,y)... a f(,) 8 (x,y) 8, where f r (,) (x,y) = p= q= βr pqx p y q are some polynomials over the field Z 7. To find them let s perform interpolation for each commutator r =,,..., 8. To find f r (,) (x,y) it is required to solve a system of linear equations over the given field: p= q= β r pqx p y q = z yx r x,y =,,,,,, (9) where zr yx is a value of r-th commutator for the pair (y,x). This system will have variables and consist of equations. 88
Alexander A.Kuznetsov, Konstantin V.Safonov Hall s Polynomials of Finite Two -Generator Groups... Let s show how to find 8 (x,y) at the example of the 8-th commutator. For short, let s write β pq instead of βpq. 8 Substituting in (9) all values of z yx 8 we receive: β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β =. The rank of the system matrix is equal to, therefore it has the only solution: β =, β =, β =, all the remaining coefficients are equal to zero. Therefore, 8 (x,y) = xy + xy + xy. Similar calculations are applied for all polynomials f r (,) (x,y). Here they are: (x,y) = x, (x,y) = y, (x,y) = xy, (x,y) = xy + x y, (x,y) = xy + xy, (x,y) = xy + x y + x y, 7 (x,y) = xy + xy + x y + x y, 8 (x,y) = xy + xy + xy. Thus, a y ax = (x, y, xy, xy + x y, xy + xy,xy + x y + x y, Using this method let s calculate other noncommutative a y j ax i. xy + xy + x y + x y,xy + xy + xy ). a y ax = (x,,y,xy,,xy + x y,,), 89
Alexander A.Kuznetsov, Konstantin V.Safonov Hall s Polynomials of Finite Two-Generator Groups... a y ax = (,x,y,,xy,,,xy + x y), a y ax = (x,,,y,,xy,,), a y ax = (,x,,y,,,xy,), a y ax = (x,,,,y,,xy,), a y ax = (,x,,,y,,,xy). Not listed pairs are commutative, i.e. a y j ax i = ax i ay j. Thus, we have a complete set of relations for the implementation of the collection process in analytical form: a y j ax i = a x i a y j af(i,j) j+ (x,y) j+ a f(i,j) j+ (x,y) 8 (x,y) j+... a f(i,j) 8, i < j 8. () Using () we can calculate the product a x...ax8 8 ay...ay8 8 = a z...az8 8. Following this procedure we will find all z i ( 8). The theorem is proved. The investigation was supported by the Ministry of Education and Science of the Russian Federation (Project Б /). References [] E.O Brien, M.Vaughan Lee, The -Generator Restricted Burnside Group of Exponent 7, Int. J. Algebra Comput., (), 9 7. [] C.Sims, Computation with finitely presented groups, Cambridge, Cambridge University Press, 99. [] D.Holt, B.Eick, E.O Brien, Handbook of computational group theory, Boca Raton, Chapman & Hall/CRC Press,. [] Ph.Hall, Nilpotent groups, Notes of lectures given at the Canadian Mathematical Congress 97 Summer Seminar, in the collected works of Philip Hall, Oxford, Clarendon Press, 988,. [] A.A.Kuznetsov, A.S.Kuznetsova, Computer modeling of finite two-generator groups of exponent five, Vestnik SibGAU, (), no., 9 (in Russian). [] A.A.Kuznetsov, A.S.Kuznetsova, Relation betweengrowth functions in symmetric groups and tasks of combinatorial optimization, Vestnik SibGAU, (), no., 7 (in Russian). [7] A.A.Kuznetsov, A.S.Kuznetsova, Fast multiplication in finite two-generated groups of exponent five, Prikl. Diskr. Mat., 8(), no., (in Russian). Полиномы Холла конечных двупорожденных групп периода семь Александр А. Кузнецов Константин В. Сафонов Пусть B k = B (, 7, k) максимальная конечная двупорожденная бернсайдова группа периода 7 ступени нильпотентности k. В настоящей статье вычислены полиномы Холла для B k при k. Ключевые слова: периодическая группа, собирательный процесс, полиномы Холла. 9