N. A. Shchuchkin, “The structure of finite semiabelian $n$-ary groups”, Chebyshevskii Sb., 17:1 (2016), 254

Abstract: The theory of

n

$n$ -ary groups emerged as a generalization of the theory of ordinary (binary) groups. Many definitions of group theory have

n

$n$ -ary analogue in the theory of

n

$n$ -ary groups. For example,

n

$n$ -ary analogs of abelian groups are abelian and semiabelian

n

$n$ -ary group.

n

$n$ -ary group

⟨ G, f ⟩

$\langle G,f \rangle$ is called semiabelian if it is true identity

f (x_{1}, x_{2}, \dots, x_{n - 1}, x_{n}) = f (x_{n}, x_{2}, \dots, x_{n - 1}, x_{1}) .

$f(x_1,x_2,\ldots,x_{n-1},x_n)=f(x_n,x_2,\ldots,x_{n-1},x_1).$
If in the

n

$n$ -ary group

⟨ G, f ⟩

$\langle G,f \rangle$ is true identities

f (x_{1}, \dots, x_{n}) = f (x_{σ (1)}, \dots, x_{σ (n)})

$f(x_1, \ldots, x_n) = f(x_{\sigma (1)}, \ldots, x_{\sigma (n)})$
for any permutation

σ \in S_{n}

$\sigma \in S_n$ , then it is called abelian.
There is a close connection between groups and

n

$n$ -ary groups. We note special case of Gluskin-Hosszu Theorem for semiabelian

n

$n$ -ary groups. On any semiabelian

n

$n$ -ary group

⟨ G, f ⟩

$\langle G,f\rangle$ it is possible to define an abelian group

⟨ G, + ⟩

$\langle G,+ \rangle$ , where

a + b = f (a, c, \dots, c, ¯ c, b)

$a+b=f (a, c,\ldots, c,\bar c, b)$ for

c

$c$ from

G

$G$ . Then for the element

d = f (c, \dots, c)

$d=f (c,\ldots, c)$ and automorphism

φ (x) = f (c, x, c, \dots, c, ¯ c)

$\varphi (x)=f (c, x, c,\ldots, c,\bar c)$ of group

⟨ G, + ⟩

$\langle G,+\rangle$ , is true equalities

φ (d) = d

$\varphi(d)=d$ ,

φ^{n - 1} (x) = x

$\varphi^{n-1}(x)=x$ for any

x \in G

$x\in G$ ,

f (a_{1}, \dots, a_{n}) = a_{1} + φ (a_{2}) + \dots + φ^{n - 2} (a_{n - 1}) + a_{n} + d .

$f(a_1,\ldots,a_n)=a_1+\varphi(a_2)+\ldots+\varphi^{n-2}(a_{n-1})+a_n+d.$
Group

⟨ G, + ⟩

$\langle G,+\rangle$ is called the retract of

n

$n$ -ary groups

⟨ G, f ⟩

$\langle G, f\rangle$ and denoted by

r e t_{c} ⟨ G, f ⟩

$ret_c\langle G,f\rangle$ . And the opposite is true: in any abelian group

⟨ G, + ⟩

$\langle G,+\rangle$ for selected automorphism

φ

$\varphi$ and element

d

$d$ with the above conditions are set semiabelian

n

$n$ -ary group

⟨ G, f ⟩

$\langle G, f\rangle$ .

n

$n$ -Ary group

⟨ G, f ⟩

$\langle G, f\rangle$ in this case, called (

φ, d

$\varphi, d$ )-derived from the group

⟨ G, + ⟩

$\langle G,+ \rangle$ and denoted by

d e r_{φ, d} ⟨ G, + ⟩

$der_{\varphi, d} \langle G,+ \rangle$ .
Let

⟨ G, f ⟩ = d e r_{φ, d} ⟨ G, + ⟩

$\langle G, f \rangle = der_{\varphi, d} \langle G, + \rangle$ – semiabelian

n

$n$ -ary group. For every automorphism

$\varphi '$ of group

$\langle G, + \rangle$ , which is conjugate to the automorphism

$\varphi$ , on the group

$\langle G, + \rangle$ we consider the endomorphism

$\mu_{\varphi'}(x)=x+\varphi'(x)+\ldots+{\varphi'}^{n-2}(x).$

$Im ~\mu_{\varphi'}$ – image of this endomorphism. Let

$\varphi'=\theta\circ\varphi\circ\theta^{-1}$ . Then, for each such automorphism

$\theta$ have coset

$\theta(d)+Im ~\mu_{\varphi'}$ of the subgroup

$Im ~\mu_{\varphi'}$ . Collection

$\{\theta(d)+Im ~\mu_{\varphi'} ~|~ \theta\in Aut ~\langle G,+\rangle \}$ all such cosets we call defining collection of sets for

$n$ -ary group

$\langle G,f\rangle$ . It is proved that semiabelian

$n$ -ary group

$\langle G,f\rangle=der_{\varphi,d}\langle G,+\rangle$ и

$\langle G,f'\rangle=der_{\psi,q}\langle G,+\rangle$ are isomorphic iff automorphisms

$\varphi$ and

$\psi$ are conjugate in group of automorphisms of group

$\langle G, +\rangle$ and defining collection of sets for these

$n$ -ary groups is equal up to permutation.
We study the finite semiabelian

$n$ -ary groups. It is shown that any semiabelian

$n$ -ary group

$\langle G, f \rangle$ of order

$|G| = p_1^{\alpha_1} p_2^{\alpha_2} \ldots p_k^{\alpha_k}$ is isomorphic to the direct product

$\langle G_1, f_1 \rangle \times \langle G_2, f_2 \rangle \times \ldots \times \langle G_k, f_k \rangle$

$n$ -ary

$p_i$ -groups

$\langle G_i, f_i \rangle$ of orders

$|G_i| = p_i^{\alpha_i}$ , where

$p_i$ – distinct primes. This decomposition is uniquely determined.
Based on the above decomposition of finite semiabelian

$n$ -ary groups into a direct product of primary semiabelian

$n$ -ary groups and for its uniqueness, we come to the main assertion about finite semiabelian

$n$ -ary groups: Any semiabelian finite

$n$ -ary group is isomorphic to the direct product of primary semiabelian

$n$ -ary groups. Any two these decompositions have the same number of factors and primary factors in these decompositions on a the same prime number have the same invariants.
It is proved the main theorem on the structure of finite abelian

$n$ -ary groups: Any finite abelian

$n$ -ary group is isomorphic to the direct product of primary abelian semicyclic

$n$ -ary groups. Any two these decompositions have the same number of factors of each order and for each prime divisor of the order of

$n$ -ary group the primary factors in these decompositions have the same invariants.
Bibliography: 18 titles.