Generating sets of the $n$-ary groups

A. M. Gal'mak ; N. A. Shchuchkin

Čebyševskij sbornik, Tome 15 (2014) no. 1, pp. 89-109.

Résumé

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 $Z(A)$.

Détail

Keywords: $n$-ary group, generatjngs sets, automorphism.

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