Definition of $ n $-ary group is obtained from the definition of group
by replacement of associative and reversible binary operation on
$ n $-ary associative operation, uniquely reversible at each site.
In this paper we study the connection between the generating sets
$ n $-ary group and
the generating sets the group to which
reducible given $ n $-ary group, according to
Post–Gluskin–Hossu theorem.
In the first part of the article describes the process that allows
knowing the generating set of the group to which this is reducible $ n $-ary group in accordance with this theorem, find a generating set of the most $ n $-ary group.
We prove that if the group $\langle A,\circ_a\rangle$,
obtained by an element $a$ of $n$-ary group $\langle A,[~]\rangle$
in accordance with Post–Gluskin–Hossu theorem, generated by a set $ M $,
then $ n $-ary group $\langle A,[~]\rangle$ generated by a set $M\cup\{a\}$.
$ n $-Ary group $\langle A,[~]\rangle$ called derived of
group $ A $, if $$[a_1a_2\ldots a_n]=a_1a_2\ldots a_n$$ for any
$a_1,a_2,\ldots, a_n\in A$. Found conditions under which
generating sets the group and $ n $-ary group, derived of this group, are identical.
We prove that the $ n $-ary group $\langle A,[~]\rangle$, derived of group $\langle A,\circ\rangle$ with identity $ e $ and generating set $ M $, is generated by a set $ M $ too, if
$$c_1\circ c_2\circ\ldots\circ c_{m(n-1)+1}=e$$ for some
$c_1,c_2,\ldots, c_{m(n-1)+1}\in M$, $m\geq 1$.
From this we deduce corollary: $ n $-ary group $\langle A,[~]\rangle$, derived of
group $\langle A,\circ\rangle$ finite period $m(n-1)+1\geq 3$
with generating set $ M $, is generated by a set $ M $ too. In
specifically, $ n $-ary group $\langle A,[~]\rangle$, derived of
cyclic group $\langle A,\circ\rangle$ of order $m(n-1)+1\geq 3$
is cyclic and is generated by the same element that
group $\langle A,\circ\rangle$.
Are a few examples of finding generating sets for $ n $-ary groups.
In the second part we study the inverse problem of finding generators
sets of binary groups, if we know the generating sets of $ n $-ary
groups from which this binary groups are obtained (according to the Post–Gluskin–Hossu theorem).
Proved that the group $\langle
A,\circ_a\rangle$, obtained by an element $ a $ of $ n $-ary
group $\langle A,[~]\rangle$ with generating set $ M $, generated by the set
$M\cup\{d=[\underbrace{a\ldots a}_n]\}$, if the automorphism
$\beta(x)=[ax\bar a\underbrace{a\ldots a}_{n-3}]$ of group $\langle
A,\circ_a\rangle$ is satisfied
\begin{equation}
M^{\beta}=\{[aM\bar a\underbrace{a\ldots a}_{n-3}]\}\subseteq M.
\label{a1'}
\end{equation}
From this we have the corollary: let $ n $-ary group $\langle A,[~]\rangle$
generated by a set $ M $, satisfying (1) for some $a\in M$. Then:
the group $\langle A,\circ_a\rangle$ generated by the set
$(M\diagdown\{a\})\cup\{d\};$
if $ a $ – idempotent in $\langle A,[~]\rangle$, then the group
$\langle A,\circ_a\rangle$ generated by the set
$M\diagdown\{a\}$.
At the end of the work described generating sets of binary groups
$\langle A,\circ_a\rangle$, found from the known generating sets of $ n $-ary
groups $\langle A,[~]\rangle$ with nonempty center $Z(A)$.
