Endomorphisms of semicyclic $n$-groups
N. A. Shchuchkin Volgograd
State Socio-Pedagogical University (Volgograd)
Abstract:
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 $f_3 (z_1, \ldots, z_n) = z_1 + mz_2 + \ldots + m ^ {n- 2} z_ {n-1} + z_n + l$,
$1 \leq m <k$ 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
$k$.
Keywords:
$n$-group, $(n,2)$-ring, $(n,2)$-nearring, endomorphism.
UDC:
512.548 Received: 12.11.2020
Accepted: 21.02.2021
DOI:
10.22405/2226-8383-2018-22-1-353-369