Hall's polynomials for finite two-generator groups of exponent seven

A. A. Kuznetsov ; K. V. Safonov

Prikladnaya Diskretnaya Matematika. Supplement, no. 7 (2014), pp. 162-164.

Résumé

Let $B_k=B_0(2,7,k)$ be the largest two-generator finite group of exponent $7$ and nilpotency class $k$. In this class, the largest group is the group $B_{28}$, which has the order $7^{20416}$. For each $B_k$, a power commutator presentation is obtained. Let $a_1^{x_1}\dots a_n^{x_n}$ and $a_1^{y_1}\dots a_n^{y_n}$ be two arbitrary elements in the group $B_k$ recorded in the commutator form. Then their product is equal $a_1^{x_1}\dots a_n^{x_n}\cdot a_1^{y_1}\dots a_n^{y_n}=a_1^{z_1}\dots a_n^{z_n}$. Powers $z_i$ are to be found based on the collection process 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 Ph. Hall. Hall showed that $z_i$ are polynomial functions (over the field $\mathbb Z_7$ in this case) depending on the variables $x_1,\dots, x_i, y_1,\dots,y_i$, which are now called Hall's polynomials. Hall's polynomials are necessary in solving problems that require multiple products of the elements of the group. Studying the structure of the Cayley graph for a group is one of these problems. The computational experiments carried out on the computer in two-generator groups of exponent five 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 for $B_k$ groups. It should be also 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 $k>4$, polynomials are calculated similarly but their output takes considerably more space so it makes impossible to verify the proof without use of computers.

Détail

Keywords: periodic group, collection process, Hall's polynomials.

Bibliographie

[1] O'Brien E. A., Vaughan-Lee M. R., “The 2-generator restricted Burnside group of exponent 7”, Int. J. Algebra Comput., 12 (2002), 459–470 | MR

[2] Sims C., Computation with Finitely Presented Groups, Cambridge University Press, Cambridge, 1994, 628 pp. | MR

[3] Holt D., Eick B., O'Brien E., Handbook of Computational Group Theory, Chapman Hall/CRC Press, Boca Raton, 2005, 514 pp. | MR | Zbl

[4] Hall P., Nilpotent groups, Notes of lectures given at the Canadian Mathematical Congress summer seminar (University of Alberta, 12–30 August, 1957), Queen Mary College, London, 1969 | Zbl

[5] Kuznetsov A. A., Kuznetsova A. S., “Kompyuternoe modelirovanie konechnykh dvuporozhdennykh grupp perioda 5”, Vestnik Sibirskogo gosudarstvennogo aerokosmicheskogo universiteta, 2012, no. 5, 59–62

[6] Kuznetsov A. A., Kuznetsova A. S., “O vzaimosvyazi funktsii rosta v simmetricheskikh gruppakh s zadachami kombinatornoi optimizatsii”, Vestnik Sibirskogo gosudarstvennogo aerokosmicheskogo universiteta, 2012, no. 6, 57–62

[7] Kuznetsov A. A., Kuznetsova A. S., “Bystroe umnozhenie elementov v konechnykh dvuporozhdënnykh gruppakh perioda pyat”, Prikladnaya diskretnaya matematika, 2013, no. 1, 110–116