Hall's polynomials over burnside groups of exponent three

Let Bk = B(k, 3) be the k-generator Burn-side group of exponent 3. Levi and van der Waerden proved that |B _k | = 3 + (2)+(3) and Bk are nilpotent of class at most 3. For each Bk a power commutator presentation can be easily obtained using the system of computer algebra GAP or MAGMA. Let a^ ... n" and a! ... an be two arbitrary elements in the group B _k recorded in the commutator form. Then their product is equal to a^ ... an · a^ ... a ™ = a1 ... an . 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 Z ₃ in this case) depending on the variables x ₁,..., x _i, y ₁,..., 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 ^ 4 are calculated within the framework of this paper. For k > 4, the polynomials are calculated similarly but their output takes considerably more space

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