DIRECT PRODUCT OF n-ARY GROUPS

The notion of n-ary group is a generalization of the binary group so many of the results from the theory of groups have n-ary analogue in theory of n-ary groups. But there are significant differences in these theories. For example, multiplier of the direct product of n-ary groups does not always have isomorphic copy in this product (in paper there is an example). It is proved that the direct product ∏ i∈I ⟨Ai , fi⟩ n-ary groups has n-ary subgroup isomorphic to ⟨Aj , fj ⟩ (j ∈ I), then and only when there is a homomorphism of ⟨Aj , fj ⟩ in ∏ i∈I,i̸=j ⟨Ai , fi⟩. Were found necessary and sufficient conditions for in direct product of n-ary groups, each of the direct factors had isomorphic copy in this product and the intersection of these copies singleton (as well as in groups) – each direct factor has a idempotent. For every n-ary group, can define a binary group which helps to study the n-ary group, that is true Gluskin-Hossu theorem: for every n-ary group of ⟨G, f⟩ for an element e ∈ G can define a binary group ⟨G, ·⟩, in which there will be an automorphism φ(x) = f(e, x, cn−2 1 ) and an element d = f( (n) e ) such that the following conditions are satisfied: f(x n 1 ) = x1 · φ(x2) · . . . · φ n−1 (xn) · d, x1, x2, . . . , xn ∈ G; (4) φ(d) = d; (5) φ n−1 (x) = d · x · d −1 , x ∈ G. (6) Group ⟨G, ·⟩, which occurs in Gluskin-Hossu theorem called retract n-ary groups ⟨G, f⟩. Converse Gluskin-Hossu theorem is also true: in any group ⟨G, ·⟩ for selected automorphism φ and element d with the terms (5) and (6), given n-ary group ⟨G, f⟩, where f defined by the rule (4). A n-ary group called (φ, d)- defined on group ⟨G, ·⟩ and denote derφ,d⟨G, ·⟩. Was found connections between n-ary group, (φ, d)-derived from the direct product of groups and n-ary groups that (φi , di)-derived on multipliers of this product: let ∏ i∈I ⟨Ai , ·i⟩ – direct product groups and φi , di – automorphism and an element in group ⟨Ai , ·i⟩ with the terms of (5) and (6) for any i ∈ I. Then derφ,d ∏ i∈I ⟨Ai , ·i⟩ = ∏ i∈I derφi,di ⟨Ai , ·i⟩, where φ – automorphism of direct product of groups ∏ i inI ⟨Ai , ·i⟩, componentwise given by the rule: for every a ∈ ∏ i∈I Ai , φ(a)(i) = φi(a(i)) (called diagonal automorphism), and d(i) = di for any i ∈ I. In the theory of n-ary groups indecomposable n-ary groups are finite primary and infinite semicyclic n-ary groups (built by Gluskin-Hossu theorem on cyclic groups). We observe n-ary analogue indecomposability cyclic groups. However, unlike groups, finitely generated semi-abelian n-ary group is not always decomposable into a direct product of a finite number of indecomposable semicyclic n-ary groups. It is proved that any finitely generated semiabelian n-ary group is isomorphic to the direct product finite number of indecomposable semicyclic n-ary groups (infinite or finite primary) if and only if in retract this n-ary group automorphism φ from Gluskin-Hossu theorem conjugate to some diagonal automorphism.

n-ary group, direct product, automorphism

1. Dornte W. Untersuchungen uber ainen verallgemeinerten Gruppenbegrief // Math. Z. Bd. 29 (1928) — P. 1–19.

2. Post E. L. Poluadic groups // Trans. Amer. Math. Soc. 48 (1940). P. 208–350.

3. Русаков С. А. Алгебраические n-арные системы. Минск: Навука i технiка, 1992.

4. Гальмак А. М. n-Арные группы. Часть I. Гомель: Гомельский гос. университет им. Ф. Скорины, 2003.

5. Курош А. Г. Общая алгебра. Лекции 1969-1970 уч. года. М.: Наука, 1974.

6. W. Dudek. A note on the axioms of n-groups / Dudek W., Glasek K., Gleichgewicht B. // Coll. Math. Soc.J.Bolyai. Vol. 29 (1977). P. 195–202.

7. Глускин Л. М. Позиционные оперативы // Мат. сборник. Т. 68(110), №3. 1965. С. 444–472.

8. Hosszu M. On the explicit form of n-group operacions // Publ. Math. 1963. Vol. 10. №1-4. P. 88-92.

9. Общая алгебра. / Под общей ред. Л. А. Скорнякова. Т. 2. М.: Наука, 1991.

10. J. Timm. Kommutative n-Gruppen. Diss„ Hamburg. 1967.

11. А. М. Гальмак. Полуабелевы n-арные группы с идемпотентами // Весник ВДУ iм П. М. Машэрава. 1999. № 2(12). С. 56–60.

12. Glasek K. and Gleichgewicht B. Abelian n-groups // Proc. Congr. Vath/ Soc. J. Bolyai. Esztergom. (1977). P. 321–329.

13. А. М. Гальмак, Г. Н. Воробьев. Тернарные группы отражений. Минск: Беларуская навука. 1998, 128 с.

14. W. A. Dudek and J. Michalski. On retrakts of polyadic groups / Dudek W.A. and Michalski J. // Demonstratio Math. 17 (1984), 281–301.

15. Щучкин Н. А. Полуциклические n-арные группы // Известия ГГУ им. Ф. Скорины. 2009. №3(54). С. 186–194.

16. Glazek K., Michalski J. and Sierocki I. On evaluation of some polyadic groups // Contributions to general algebra 3: Verlag H¨older-Pichler-Tempsky. Wiena, 1985. P. 157–171.

17. Курош А. Г. Теория групп. 3-е изд. М.: Наука, 1967. 18. В. С. Монахов. Введение в теорию конечных групп и их классов. Минск: Вышейшая школа, 2006.