The notion of $n$-ary group is a generalization of the binary group
so many of the results from the theory of groups have $n$-ary analogue in
theory of $n$-ary groups. But there are significant differences in these
theories. For example, multiplier of the direct product of $n$-ary groups
does not always have isomorphic copy in this product (in paper there is an example).
It is proved that the direct product $\prod_{i\in
I}\langle A_i,f_i\rangle$ $n$-ary groups has $n$-ary
subgroup isomorphic to $\langle A_j,f_j\rangle$ ($j\in I$), then and
only when there is a homomorphism of $\langle
A_j,f_j\rangle$ in $\prod_{i\in I,i\ne j}\langle A_i,f_i\rangle.$
Were found necessary and sufficient conditions for
in direct product of $n$-ary groups, each of the direct factors had
isomorphic copy in this product and the intersection of these copies
singleton (as well as in groups) — each direct factor has a
idempotent.
For every $n$-ary group, can define a binary group
which helps to study the $n$-ary group, that is true
Gluskin–Hossu theorem: for every $n$-ary group of $\langle
G,f\rangle$ for an element $e\in G$ can define a binary group
$\langle G,\cdot\rangle$, in which there will be an automorphism
$\varphi(x)=f(e,x,c_1^{n-2})$ and an element $d=f(\overset{(n)}{e})$
such that the following conditions are satisfied:
\begin{align*}
(x_1^n)=x_1\cdot\varphi(x_2)\cdot\ldots\cdot\varphi^{n-1}(x_n)\cdot d, ~~ x_1,x_2,\ldots,x_n\in G;\qquad\qquad\qquad\!(4)\\
\varphi(d)=d;\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\,\,\,(5)\\
\varphi^{n-1}(x)=d\cdot x\cdot d^{-1}, ~~ x\in G.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(6)
\end{align*}
Group $\langle G,\cdot\rangle$, which occurs in Gluskin–Hossu theorem
called retract $n$-ary groups $\langle G,f\rangle$.
Converse Gluskin–Hossu theorem is also true: in any group $\langle
G,\cdot \rangle $ for selected automorphism $\varphi $ and element
$d$ with the terms (5) and (6), given $n$-ary group
$\langle G,f \rangle $, where $ f $ defined by the rule (4).
A $ n $-ary group called ($\varphi, d $)-defined on
group $\langle G, \cdot \rangle $ and denote
$der_{\varphi, d} \langle G, \cdot \rangle $.
Was found connections between $n$-ary group, ($\varphi, d$)-derived
from the direct product of groups and $n$-ary groups that
($\varphi_i, d_i $)-derived on multipliers of this product:
let $\prod_{i \in I} \langle A_i, \cdot_i \rangle$ — direct
product groups and $\varphi_i$, $d_i$ — automorphism and an element in
group $\langle A_i, \cdot_i \rangle$ with the terms of (5) and
(6) for any $i \in I$. Then
$$der_{\varphi, d} \prod_{i \in I} \langle A_i, \cdot_i \rangle = \prod_{i \in I} der_{\varphi_i, d_i} \langle
A_i, \cdot_i \rangle,$$ where $\varphi$ – automorphism of direct
product of groups $\prod_{i \ in I} \langle A_i, \cdot_i \rangle $,
componentwise given by the rule: for every $a \in \prod_{i \in
I} A_i $, $ \varphi (a) (i) = \varphi_i (a (i)) $ (called diagonal automorphism),
and $d (i) = d_i $ for any $i \in I$.
In the theory of $n$-ary groups indecomposable $n$-ary groups
are finite primary and infinite semicyclic
$n$-ary groups (built by Gluskin–Hossu theorem on
cyclic groups). We observe $n$-ary analogue indecomposability
cyclic groups. However, unlike groups, finitely generated
semi-abelian $n$-ary group is not always decomposable into a direct
product of a finite number of indecomposable semicyclic
$n$-ary groups. It is proved that any finitely generated
semi-abelian $n$-ary group is isomorphic to the direct product
finite number of indecomposable semicyclic $n$-ary groups
(infinite or finite primary) if and only if in
retract this $n$-ary group automorphism $\varphi$ from
Gluskin–Hossu theorem conjugate to some diagonal automorphism.
Bibliography: 18 titles.
