THE STRUCTURE OF FINITE SEMIABELIAN n-ARY GROUPS

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 ⟨G, f⟩ is called semiabelian if it is true identity f(x1, x2, . . . , xn−1, xn) = f(xn, x2, . . . , xn−1, x1). If in the n-ary group ⟨G, f⟩ is true identities f(x1, . . . , xn) = f(xσ(1), . . . , xσ(n)) for any permutation σ ∈ Sn, 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 ⟨G, f⟩ it is possible to define an abelian group ⟨G,+⟩, where a + b = f(a, c, . . . , c, ¯c, b) for c from G. Then for the element d = f(c, . . . , c) and automorphism ϕ(x) = f(c, x, c, . . . , c, ¯c) of group ⟨G,+⟩, is true equalities ϕ(d) = d, ϕn−1(x) = x for any x ∈ G, f(a1, . . . , an) = a1 + ϕ(a2) + . . . + ϕn−2(an−1) + an + d. Group ⟨G,+⟩ is called the retract of n-ary groups ⟨G, f⟩ and denoted by retc⟨G, f⟩. And the opposite is true: in any abelian group⟨G,+⟩ for selected automorphism ϕ and element d with the above conditions are set semiabelian n-ary group ⟨G, f⟩. n-Ary group ⟨G, f⟩ in this case, called (ϕ, d)-derived from the group ⟨G,+⟩ and denoted by derϕ,d⟨G,+⟩. Let ⟨G, f⟩ = derϕ,d⟨G,+⟩ – semiabelian n-ary group. For every automorphism ϕ′ of group ⟨G,+⟩, which is conjugate to the automorphism ϕ, on the group ⟨G,+⟩ we consider the endomorphism μϕ′ (x) = x+ϕ′(x)+. . .+ϕ′n−2(x). Im μϕ′ – image of this endomorphism. Let ϕ′ = θ ◦ϕ◦ θ−1. Then, for each such automorphism θ have coset θ(d)+Im μϕ′ of the subgroup Im μϕ′ . Collection {θ(d) + Im μϕ′ | θ ∈ Aut ⟨G,+⟩} all such cosets we call defining collection of sets for n-ary group ⟨G, f⟩. It is proved that semiabelian n-ary group ⟨G, f⟩ = derϕ,d⟨G,+⟩ и ⟨G, f′⟩ = derψ,q⟨G,+⟩ are isomorphic iff automorphisms ϕ and ψ are conjugate in group of automorphisms of group ⟨G,+⟩ 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 ⟨G, f⟩ of order |G| = pα1 1 pα2 2 . . . pαk k is isomorphic to the direct product ⟨G1, f1⟩ × ⟨G2, f2⟩ × . . . × ⟨Gk, fk⟩ n-ary pi-groups ⟨Gi, fi⟩ of orders |Gi| = pαi i , where pi – 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.

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