ALGORITHM FOR CALCULATION IN SYLOW 2-SUBGROUPS OF ALTERNATING GROUPS USING THE COMPUTER ALGEBRA SYSTEM GAP

The Sylow 2-subgroups of symmetric groups was described by Leo Kaluzhnin. He presented the elements of these groups as a tables, i.e. the ordered sets of polynomials of a certain form. The Sylow 2-subgroups of symmetric groups was studied by V. Sushchanskii, Yu. Dmytruk, A. Slupik and other mathematicians. In this paper the Sylow 2-subgroups of alternating groups are characterized. The system of computer algebra GAP was used for this characterization. System of computer algebra GAP is the most popular frequency of references in scientific publications and the number of links on Internet pages. Its popularity is conditioned by accessibility, large set of functions and packages for calculations in theoretical and mathematical sciences, clarity and ease to use. At the moment, it is an auxiliary tool for working with groups, finite fields, algebraic extension of fields, Galois group, polynomials of many variables, rational functions, vectors, matrices, etc. In addition, this list is supplemented every day. The CMG-program (Check Minimal Generators) is provided in this article. It is created using the GAP system and is used for calculations in Sylow 2-subgroups of alternating groups. The main task of the program is to check whether some set S can be a system of generators of the Sylow 2-subgroup of alternating group. The program are used for groups Syl2(A8) and Syl2(A16). In addition, the commutator and the factor subgroup of these groups are investigated. It is shown that each element of the commutator subgroup of the group Syl2(A8) is commutator in this group. Moreover, the subgroup of order 4 of the commutator of the group Syl2(A8) is described. Also, this program checks whether the commutators and the factor groups of groups Syl2(A8) and Syl2(A16) are Abelian.