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.
Bibliography: 18 titles.
