Endomorphisms of semicyclic $n$-groups

N. A. Shchuchkin

Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 353-369.

Résumé

One of the main problems for semiabelian $n$-groups is the finding of $(n,2)$-nearrings, which are isomorphic to $(n,2)$-nearrings of endomorphisms of certain semiabelian $n$-groups. Such almost $(n,2)$-nearrings are found for semicyclic $n$-groups. 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 $Z$ we select the set $P = \{m | ml \equiv l \pmod {n-1} \} $ and define an $ n$-ary operation $h$ by the rule $h (m_1, \ldots, m_n) = m_1 + \ldots + m_n$ on this set. Then the algebra $\langle P, h, \cdot \rangle $, where $\cdot $ is the multiplication of integers, is a $(n,2)$-ring. It is proved that $\langle P,h,\cdot\rangle$ is isomorphic to $(n,2)$-ring of endomorphisms of semicyclic $n$-group of type $(\infty,1, l)$. In the $n$-group $\langle Z \times Z, h \rangle = \langle Z, f_2 \rangle \times \langle Z, f_2 \rangle $ we define the binary operation $\diamond $ by the rule $(m_1, u_1) \diamond (m_2, u_2) = (m_1m_2, m_1u_2 + u_1).$ Then $\langle Z \times Z, h, \diamond \rangle $ is an $(n, 2)$-nearringsg. It is proved that $\langle Z \times Z, h, \diamond \rangle $ is isomorphic to $(n, 2)$-nearrings of endomorphisms of a semicyclic $n$-group of type $(\infty, -1,0)$. It is proved that $(n, 2)$-ring $\langle Z, f, * \rangle $, where $f (z_1, \ldots, z_n) = z_1 + \ldots + z_n + 1$ and $z_1 * z_2 = z_1z_2 (n-1) + z_1 + z_2$, is isomorphic to $(n, 2)$-rings of endomorphisms of infinite cyclic $n$-group. On additive group of the ring of residue classes of $Z_k$ we define $n$-group $\langle Z_k, f_3 \rangle$, where the $n$-ary operation $f_3$ operates according to the rule $f_3 (z_1, \ldots, z_n) = z_1 + mz_2 + \ldots + m ^ {n- 2} z_ {n-1} + z_n + l$, $1 \leq m $ and $m$ is relatively prime to $k$. In addition, $m$ satisfies the congruence $lm \equiv l \pmod {k}$ and multiplicative order of $m$ modulo $k$ divides $n-1$. Any finite semicyclic $n$-group of order $ k$ is isomorphic to $n$-group $\langle Z_k, f_3 \rangle $, where $l \mid \mathrm{gcd} (n-1, k)$ for $m = 1$ and $l \mid \mathrm{gcd} (\frac {m ^ {n-1} -1} {m-1}, k)$ for $m \ne 1$. We will say that such $n$-group has type $(k, m, l)$. In the $n$-group $\langle P, h \rangle = \langle Z_k, f_3 \rangle \times \langle Z_l, f_4 \rangle$, $f_4 (z_1, \ldots, z_n) = z_1 + rz_2 + \ldots + r ^ {n-2} z_ {n-1} + z_n$, where $r$ is the remainder of dividing $m$ by $l$, we define the binary operation $\diamond$ by the rule $$ (u_1, v_1) \diamond (u_2, v_2) = (u_2s_1 + u_1, v_2s_1 + v_1)$$ where $s_1 \in Z_k$, $s_1-1 = s_0 + v_1 \frac {k} {l}$, and $s_0$ is solution of congruence $x \equiv \frac {(n-1) u_1} {l}\pmod {\frac {k} {l}}$ for $m = 1$ and $x \equiv \frac {\frac {m ^ {n-1} -1} {m-1} u_1} {l} \pmod {\frac {k} {l}}$ for $m \ne 1 $. It is proved that the algebra $\langle P, h, \diamond \rangle$ is $(n, 2)$-ring for $m = 1$ and $(n, 2)$-nearring for $m \ne 1$, which is isomorphic to $(n, 2)$-ring of endomorphisms of abelian semicyclic $n$-group of type $(k, 1, l)$ with $m = 1$ and $(n, 2)$-nearring of endomorphisms of semicyclic $n$-groups of type $(k, m, l)$ for $m \ne 1$. It is proved that $(n, 2)$-ring $\langle Z_k, f, * \rangle$, where $f (z_1, \ldots, z_n) = z_1 + \ldots + z_n + 1$ and $u_1 * u_2 = u_1 \cdot u_2 \cdot (n-1) + u_1 + u_2$, is isomorphic to $(n, 2)$-ring of endomorphisms of finite cyclic $n$-group of order $k$.

Détail

Keywords: $n$-group, $(n,2)$-ring, $(n,2)$-nearring, endomorphism.

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