Direct product of $n$-ary groups

N. A. Shchuchkin

Čebyševskij sbornik, Tome 15 (2014) no. 2, pp. 101-121.

Résumé

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 $\prod_{i\in I}\langle A_i,f_i\rangle$ $n$-ary groups has $n$-ary subgroup isomorphic to $\langle A_j,f_j\rangle$ ($j\in I$), then and only when there is a homomorphism of $\langle A_j,f_j\rangle$ in $\prod_{i\in I,i\ne j}\langle A_i,f_i\rangle.$ 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 $\langle G,f\rangle$ for an element $e\in G$ can define a binary group $\langle G,\cdot\rangle$, in which there will be an automorphism $\varphi(x)=f(e,x,c_1^{n-2})$ and an element $d=f(\overset{(n)}{e})$ such that the following conditions are satisfied: \begin{align*} (x_1^n)=x_1\cdot\varphi(x_2)\cdot\ldots\cdot\varphi^{n-1}(x_n)\cdot d, ~~ x_1,x_2,\ldots,x_n\in G;\qquad\qquad\qquad\!(4)\\ \varphi(d)=d;\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\,\,\,(5)\\ \varphi^{n-1}(x)=d\cdot x\cdot d^{-1}, ~~ x\in G.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(6) \end{align*} Group $\langle G,\cdot\rangle$, which occurs in Gluskin–Hossu theorem called retract $n$-ary groups $\langle G,f\rangle$. Converse Gluskin–Hossu theorem is also true: in any group $\langle G,\cdot \rangle $ for selected automorphism $\varphi $ and element $d$ with the terms (5) and (6), given $n$-ary group $\langle G,f \rangle $, where $ f $ defined by the rule (4). A $ n $-ary group called ($\varphi, d $)-defined on group $\langle G, \cdot \rangle $ and denote $der_{\varphi, d} \langle G, \cdot \rangle $. Was found connections between $n$-ary group, ($\varphi, d$)-derived from the direct product of groups and $n$-ary groups that ($\varphi_i, d_i $)-derived on multipliers of this product: let $\prod_{i \in I} \langle A_i, \cdot_i \rangle$ — direct product groups and $\varphi_i$, $d_i$ — automorphism and an element in group $\langle A_i, \cdot_i \rangle$ with the terms of (5) and (6) for any $i \in I$. Then $$der_{\varphi, d} \prod_{i \in I} \langle A_i, \cdot_i \rangle = \prod_{i \in I} der_{\varphi_i, d_i} \langle A_i, \cdot_i \rangle,$$ where $\varphi$ – automorphism of direct product of groups $\prod_{i \ in I} \langle A_i, \cdot_i \rangle $, componentwise given by the rule: for every $a \in \prod_{i \in I} A_i $, $ \varphi (a) (i) = \varphi_i (a (i)) $ (called diagonal automorphism), and $d (i) = d_i $ for any $i \in I$. 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 semi-abelian $n$-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 $\varphi$ from Gluskin–Hossu theorem conjugate to some diagonal automorphism. Bibliography: 18 titles.

Détail

Keywords: $n$-ary group, direct product, automorphism.

Bibliographie

[1] Dornte W., “Untersuchungen uber ainen verallgemeinerten Gruppenbegrief”, Math. Z., 29 (1928), 1–19 | DOI | MR | Zbl

[2] Post E. L., “Poluadic groups”, Trans. Amer. Math. Soc., 48 (1940), 208–350 | DOI | MR

[3] Rusakov S. A., Algebraicheskie $n$-arnye sistemy, Navuka i tekhnika, Minsk, 1992

[4] Galmak A. M., $n$-Arnye gruppy, v. I, Gomelskii gos. universitet im. F. Skoriny, Gomel, 2003

[5] Kurosh A. G., Obschaya algebra. Lektsii 1969–1970 uch. goda, Nauka, M., 1974 | MR

[6] Dudek W., Glasek K., Gleichgewicht B., “A note on the axioms of $n$-groups”, Coll. Math. Soc. J. Bolyai, 29, 1977, 195–202 | MR

[7] Gluskin L. M., “Pozitsionnye operativy”, Mat. sbornik, 68(110):3 (1965), 444–472 | MR | Zbl

[8] Hosszu M., “On the explicit form of $n$-group operacions”, Publ. Math., 10:1–4 (1963), 88–92 | MR

[9] L. A. Skornyakov (red.), Obschaya algebra, v. 2, Nauka, M., 1991 | Zbl

[10] Timm J., Kommutative $n$-Gruppen, Diss., Hamburg, 1967

[11] Galmak A. M., “Poluabelevy $n$-arnye gruppy s idempotentami”, Vesnik VDU im P. M. Masherava, 1999, no. 2(12), 56–60

[12] Glasek K., Gleichgewicht B., “Abelian $n$-groups”, Proc. Congr. Vath. Soc. J. Bolyai (Esztergom, 1977), 321–329 | MR

[13] Galmak A. M., Vorobev G. N., Ternarnye gruppy otrazhenii, Belaruskaya navuka, Minsk, 1998, 128 pp.

[14] Dudek W. A., Michalski J., “On retrakts of polyadic groups”, Demonstratio Math., 17 (1984), 281–301 | MR | Zbl

[15] Schuchkin N. A., “Polutsiklicheskie $n$-arnye gruppy”, Izvestiya GGU im. F. Skoriny, 2009, no. 3(54), 186–194

[16] Glazek K., Michalski J., Sierocki I., “On evaluation of some polyadic groups”, Contributions to general algebra 3, Verlag Hölder-Pichler-Tempsky, Wiena, 1985, 157–171 | MR

[17] Kurosh A. G., Teoriya grupp, 3-e izd., Nauka, M., 1967 | MR | Zbl

[18] Monakhov V. S., Vvedenie v teoriyu konechnykh grupp i ikh klassov, Vysheishaya shkola, Minsk, 2006