Алгоритм обчислень у силовських 2-підгрупах знакозмінних груп за допомогою системи комп'ютерної алгебри GAP
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Date
2018
Authors
Ольшевська, Віта
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Abstract
У статтi наведено алгоритм перевiрки, чи є певна множина елементiв S мiнiмальною системою
твiрних для силовської 2-пiдгрупи знакозмiнної групи Syl2(A2n ), за допомогою системи комп’ютерної
алгебри GAP. Для невеликих n (n = 3 i n = 4) проведено обчислення за допомогою цього алгоритму.
Зокрема, перевiрено абелевiсть, пораховано потужнiсть та кiлькiсть елементiв мiнiмальної системи
твiрних комутантiв у кожнiй з груп Syl2(A8), Syl2(A16) та фактор-групах цих силовських 2-пiдгруп по
комутанту.
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.
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.
Description
Keywords
групи, силовськi пiдгрупи, пiдстановки, стаття, groups, sylow subgroups, permutations, GAP
Citation
Ольшевська В. А. Алгоритм обчислень у силовських 2-підгрупах знакозмінних груп за допомогою системи комп'ютерної алгебри GAP / Ольшевська В. А. // Могилянський математичний журнал : науковий журнал. - 2018. - Т. 1. - С. 30-33.