Hall's polynomials over Burnside groups of exponent three

A. A. Kuznetsov ; K. V. Safonov

Prikladnaya Diskretnaya Matematika. Supplement, no. 8 (2015), pp. 147-149.

Résumé

Let $B_k=B(k,3)$ be the $k$-generator Burnside group of exponent 3. Levi and van der Waerden proved that $|B_k|=3^{k+\binom k2+\binom k3}$ and $B_k$ are nilpotent of class at most 3. For each $B_k$ a power commutator presentation can be easily obtained using the system of computer algebra GAP or MAGMA. 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 to $a_1^{x_1}\dots a_n^{x_n}\cdot a_1^{y_1}\dots a_n^{y_n}=a_1^{z_1}\dots a_n^{z_n}$. 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 $\mathbb Z_3$ in this case) depending on the variables $x_1,\dots,x_i,y_1,\dots,y_i$ and now called Hall's polynomials. Hall’s polynomials are necessary in solving problems, which require multiple products of the elements of the group. The study of the Cayley graph structure for a group is one of these problems. The computational experiments carried out on the computer in groups of exponent five and seven 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 more 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 $B_k$ for $k\leq4$ are calculated within the framework of this paper. For $k>4$, the polynomials are calculated similarly but their output takes considerably more space.

Détail

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

Bibliographie

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[6] Kuznetsov A. A. Safonov K. V., “Hall's polynomials of finite two-generator groups of exponent seven”, Zhurnal SFU. Ser. matematika i fizika, 7:2 (2014), 186–190