Homomorphisms from infinite semilcyclic $n$-groups to a semiabelian $n$-group

N. A. Shchuchkin

Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 340-352.

Résumé

One of the main problems for semiabelian $n$-groups is the finding of semiabelian $n$-groups, which are isomorphic to $(n)$-groups of homomorphisms from certain $n$-groups to a semiabelian $n$-group. Such $n$-groups are found for infinite semicyclic $n$-groups. It is known that the set $Hom (G, C)$ of all homomorphisms from $n$-groups $\langle G, f_1 \rangle$ to a semiabelian (abelian) $n$-group $\langle C, f_2 \rangle$ with $n$-ary operation $g$ given by the rule $$g (\varphi_1, \ldots, \varphi_n) (x) = f_2 (\varphi_1 (x), \ldots, \varphi_n (x)), x \in G,$$ forms a semiabelian (abelian) $n$-group. It is proved that the isomorphisms $\psi_1$ of $n$-groups $\langle G, f_1 \rangle$ and $\langle G ', f'_1 \rangle$ and $\ psi_2$ of semiabelian $n$-groups $\langle C , f_2 \rangle$ and $\langle C ', f'_2 \rangle$ induce an isomorphism $\tau$ of $n$-groups of homomorphisms $\langle Hom (G, C), g\rangle$ and $\langle Hom (G ', C'), g '\rangle$, which acts according to the rule $\tau: \alpha \to \psi_2 \circ \alpha \circ \psi_1 ^ {- 1}$. On the additive group of integers $Z$ we construct an abelian $n$-group $\langle Z,f_1 \rangle$ with $n$-ary operation $f_1 (z_1, \ldots, z_n) = z_1 + \ldots + z_n + l$, where $l$ is any integer. For a nonidentical automorphism $ \varphi (z) = - z$ on $Z$, we can specify semiabelian $n$-group $\langle Z, f_2 \rangle$ for $n = 2k + 1 $, $ k \in N$, with the $n$-ary operation $f_2 (z_1, \ldots, z_n) = z_1-z_2 + \ldots + z_ {2k-1} -z_ {2k} + z_ {2k + 1}$. Any infinite semicyclic $n$-group is isomorphic to either the $n$-group $\langle Z, f_1 \rangle$, where $0 \leq l \leq [\frac {n-1} {2}]$, or the $n$-group $\langle Z, f_2 \rangle$ for odd $n$. In the first case we will say that such $n$-group has type $ (\infty, 1, l)$, and in the second case, it has type $(\infty, -1,0)$. In studying the $n$-groups of homomorphisms $\langle Hom (Z, C), g \rangle$ from an infinite abelian semicyclic $n$-group $\langle Z, f_1 \rangle$ ($0 \leq l \leq \frac {n-1} {2} $) to a semiabelian $n$-group $\langle C, f_2 \rangle$ we construct on the $n$-group $\langle C, f_2 \rangle$ an abelian group $C$ with the addition operation $a + b = f_2 (a, \overset {(n-3)} {c}, \bar c, b)$, in which there is an element $ d_2 = f_2 (\overset {(n)} {c})$ and an automorphism $\varphi_2 (x) = f_2 (c, x, \overset {(n-3)} {c}, \bar c )$. Choose a set $P_1$ of such ordered pairs $(a, u)$ of elements from $C$ that satisfy the equality $ la = d_2 + \overset {\sim} {\varphi_2} (u)$, where $\overset {\sim} {\varphi_2} (x) = x + \varphi_2 (x) + \ldots + \varphi ^ {n-2} _2 (x), x \in C$ is an endomorphism of the group $C$, and for the first component of these pairs the equality is true $\varphi_2 (a) = a$. On this set, we define a $n$-ary operation $h_1$ by the rule $h_1 ((a_1, u_1), \ldots, (a_n, u_n)) = (a_1 + \ldots + a_n, f_2 (u_1, \ldots, u_n) )$. It is proved that $\langle P_1, h_1 \rangle$ is a semiabelian $n$-group, which is isomorphic to the $n$-group of homomorphisms from an infinite abelian semicyclic $n$-group $\langle Z, f_1 \rangle$ ($ 0 \leq l \leq \frac {n-1} {2}$) to an $n$-group $\langle C, f_2 \rangle$. The consequence of this isomorphism is an isomorphism of $n$-groups of $\langle P_1, h_1 \rangle$ and $n$-groups of homomorphisms from an infinite abelian semicyclic $n$-group of type $(\infty, 1, l)$ to a semiabelian $n$-group $\langle C, f_2 \rangle$. When studying the $n$-group of homomorphisms $\langle Hom (Z, C), g \rangle$ from the infinite semicyclic $n$-group $\langle Z, f'_1 \rangle$ to the semiabelian $n$-group $\langle C, f_2 \rangle$ in the abelian group $C$ choose the subgroup $H = \{a \in C ~ | ~ \varphi_2 (a) = - a \}$. On $H$ we define a semiabelian $n$-group $\langle H, h \rangle$, where $h$ acts according to the rule $h (a_1, a_2, \ldots, a_ {n-1}, a_n) = a_1 + \varphi_2 (a_2) + \ldots + \varphi ^ {n-2} _2 (a_ {n-1}) + a_n$. Then in the $n$-group $\langle C, f_2 \rangle$ we select the subgroup $\langle T, f_2 \rangle$ of all idempotents, if $T \ne \emptyset$. It is proved that for an odd number $n> 1$ a direct product of semiabelian $n$-groups $\langle H, h \rangle \times \langle T, f_2 \rangle$ is isomorphic to $n$-group of homomorphisms from infinite semicyclic $n$-groups of $\langle Z, f'_1 \rangle$ to a semiabelian $n$-group $\langle C, f_2 \rangle$ with a non empty set of idempotents $T$. The consequence of this isomorphism is the isomorphism of the $n$-group $\langle H, h \rangle \times \langle T, f_2 \rangle$ and $n$-groups of homomorphisms from an infinite semicyclic $n$-group of type $(\infty, - 1,0)$ to the semiabelian $n$-group $\langle C, f_2 \rangle$. Similar facts were obtained when studying the $n$-group of homomorphisms $\langle Hom (Z, C), g \rangle$ from $n$-groups $\langle Z, f_1 \rangle $ and $\langle Z, f'_1 \rangle$ to an abelian $n$-group $\langle C, f_2 \rangle$.

Détail

Keywords: $n$-group, semiabelian $(n,2)$-group, abelian $(n,2)$-group, homomorphism.

Bibliographie

[1] Aliev I. Sh., “O naimenshem mnogoobrazii simmetricheskikh algebr”, Algebra i logika (seminar), 5:6 (1966), 5–14 | Zbl

[2] Chakani B., “Ob abelevykh svoistvakh primitivnykh klassov universalnykh algebr”, Acta scient. math., 25 (1964), 202–208

[3] Glazek K., Gleichgewicht B., “Abelian n-groups”, Proc. Congr. Math. Soc. J. Bolyai Esztergom (Hungaru), 29 (1977), 321–329 | MR

[4] Kurosh A. G., Obschaya algebra. Lektsii 1969–1970 uch., Nauka, M., 1974, 158 pp. | MR

[5] Rusakov S. A., Algebraicheskie $n$-arnye sistemy, Navuka i tekhnika, Minsk, 1992, 263 pp.

[6] Galmak A. M., $n$-Arnye gruppy, v. 1, Gomelskii gos. universitet im. F. Skoriny, Gomel, 2003, 195 pp.

[7] Galmak A. M., $n$-Arnye gruppy, v. 2, Izdatelskii tsentr BGU, Minsk, 2007, 323 pp.

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

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

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

[11] Schuchkin N. A., “Pryamoe proizvedenie $n$-arnykh grupp”, Chebyshevskii sbornik, 15:2 (2014), 101–121

[12] Glazek K., Michalski J., Sierocki A. I., “On evaluation of some polyadic groups”, Contributions to General Algebra, 3 (1985), 159–171 | MR

[13] Khodabandeh H., Shahryari M., “On the representations and automorphisms of polyadic groups”, Commun. Algebra, 40 (2012), 2199–2212 | DOI | MR | Zbl

[14] Schuchkin N. A., “Polutsiklicheskie $n$-arnye gruppy”, Izvestiya GGU im. F. Skoriny, 3(54) (2009), 186–194

[15] Fuks L., Beskonechnye abelevy gruppy, v. 1, “Mir”, M., 1974, 335 pp.

[16] Schuchkin N. A., Vvedenie v teoriyu $n$-grupp, OOO “Print”, Volgograd, 2019, 234 pp.

[17] Galmak A. M., $n$-Arnaya podgruppa edinits, Preprinty Gomelskogo gos. un-ta, 1998, 23 pp. | Zbl