GENERATING SETS OF THE N-ARY GROUPS

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 hA, ◦ai, obtained by an element a of n-ary group hA, [ ]i in accordance with Post-Gluskin-Hossu theorem, generated by a set M, then n-ary group hA, [ ]i generated by a set M ∪ {a}. n-Ary group hA, [ ]i called derived of group A, if [a1a2 . . . an] = a1a2 . . . an for any a1, a2, . . . , an ∈ 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 hA, [ ]i, derived of group hA, ◦i with identity e and generating set M, is generated by a set M too, if c1 ◦ c2 ◦ . . . ◦ cm(n−1)+1 = e for some c1, c2, . . . , cm(n−1)+1 ∈ M, m > 1. From this we deduce corollary: n-ary group hA, [ ]i, derived of group hA, ◦i finite period m(n−1)+ 1 > 3 with generating set M, is generated by a set M too. In specifically, n-ary group hA, [ ]i, derived of cyclic group hA, ◦i of order m(n − 1) + 1 > 3 is cyclic and is generated by the same element that group hA, ◦i. 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 hA, ◦ai, obtained by an element a of n-ary group hA, [ ]i with generating set M, generated by the set M ∪ {d = [a . . . a | {z } n ]}, if the automorphism β(x) = [axa a . . . a ¯ | {z } n−3 ] of group hA, ◦ai is satisfied Mβ = {[aMa a . . . a ¯ | {z } n−3 ]} ⊆ M. (2) From this we have the corollary: let n-ary group hA, [ ]i generated by a set M, satisfying (2) for some a ∈ M. Then: 1) the group hA, ◦ai generated by the set (M{a}) ∪ {d}; 2) if a – idempotent in hA, [ ]i, then the group hA, ◦ai generated by the set M{a}. At the end of the work described generating sets of binary groups hA, ◦ai, found from the known generating sets of n-ary groups hA, [ ]i with nonempty center Z(A). 

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