The structure of finite semiabelian $n$-ary groups

The theory of $n$-ary groups emerged as a generalization of the theory of ordinary (binary) groups. Many definitions of group theory have $n$-ary analogue in the theory of $n$-ary groups. For example, $n$-ary analogs of abelian groups are abelian and semiabelian $n$-ary group. $n$-ary group $\langle G,f \rangle$ is called semiabelian if it is true identity
$$f(x_1,x_2,\ldots,x_{n-1},x_n)=f(x_n,x_2,\ldots,x_{n-1},x_1).$$
If in the $n$-ary group $\langle G,f \rangle$ is true identities
$$f(x_1, \ldots, x_n) = f(x_{\sigma (1)}, \ldots, x_{\sigma (n)})$$
for any permutation $\sigma \in S_n$, then it is called abelian.
There is a close connection between groups and $n$-ary groups. We note special case of Gluskin-Hosszu Theorem for semiabelian $n$-ary groups. On any semiabelian $ n $-ary group $\langle G,f\rangle$ it is possible to define an abelian group $\langle G,+ \rangle$, where $a+b=f (a, c,\ldots, c,\bar c, b) $ for $c$ from $G$. Then for the element $d=f (c,\ldots, c)$ and automorphism $\varphi (x)=f (c, x, c,\ldots, c,\bar c) $ of group $\langle G,+\rangle$, is true equalities $\varphi(d)=d$, $\varphi^{n-1}(x)=x$ for any $x\in G$,

$$f(a_1,\ldots,a_n)=a_1+\varphi(a_2)+\ldots+\varphi^{n-2}(a_{n-1})+a_n+d.$$
Group $\langle G,+\rangle$ is called the retract of $n$-ary groups $\langle G, f\rangle$ and denoted by $ret_c\langle G,f\rangle$. And the opposite is true: in any abelian group$\langle G,+\rangle$ for selected automorphism $\varphi$ and element $d$ with the above conditions are set semiabelian $n$-ary group $\langle G, f\rangle$. $n$-Ary group $\langle G, f\rangle$ in this case, called ($\varphi, d$)-derived from the group $\langle G,+ \rangle$ and denoted by $der_{\varphi, d} \langle G,+ \rangle$.
Let $\langle G, f \rangle = der_{\varphi, d} \langle G, + \rangle $ – semiabelian $n$-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.
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