TO THE POST’S COSET THEOREM

Abstract

In the theory of polyadic groups plays an important role groups A∗ and A0, appearing in Post’s Coset Theorem [2], asserts that for every n-ary groups ⟨A, [ ]⟩ exists a group of A∗ , in which there is normal subgroup A0 such that the factor group A∗/A0 — cyclic group of order n−1. Generator xA0 this cyclic group is the n-ary group with n-ary operation derived from operation in the group A∗ , wherein n-ary groups ⟨A, [ ]⟩ and ⟨xA0, [ ]⟩ isomorphic. Group A∗ is called the Post’s universal covering group, and the group A0 — appropriate group. The article begins with a generalization of the Post’s Coset Theorem: for every n-ary groups ⟨A, [ ]⟩, n = k(m − 1) + 1, the Post’s universal covering group A∗ has a normal subgroup mA such that the factor group A∗/ mA — cyclic group of order m − 1. Moreover, A0 ⊆ mA ⊆ A∗ and mA/A0 - cyclic group of order k. In this paper we study the permutability of elements in n-ary group. In particular, we study the m-semi-commutativity in n-ary groups, which is a generalization of of the well-known concepts of commutativity and semicommutativity. Recall that the n-ary group ⟨A, [ ]⟩ is called abelian if it contains any substitution σ of the set {1, 2, . . . , n} true identity [a1a2 . . . an] = [aσ(1)aσ(2) . . . aσ(n) ], and n-ary group ⟨A, [ ]⟩ is called a semi-abelian if it true identity [aa1 . . . an−2b] = [ba1 . . . an−2a] Summarizing these two definitions, E. Post called n-ary group ⟨A, [ ]⟩ m-semiabelian if m − 1 divides n − 1 and (aa1 . . . am−2b, ba1 . . . am−2a) ∈ θA for any a, a1, . . . , am−2, b ∈ A. We have established a new criterion of m-semi-commutativity of n-ary group, formulated by a subgroup mA of the Post’s universal covering group: n-ary group ⟨A, [ ]⟩ is m-semi-abelian if and only if the group mA is abelian. For n = k(m−1) + 1 by fixed elements c1, . . . , cm−2 ∈ A on n-ary group of ⟨A, [ ]⟩ construct (k + 1)-ary group ⟨A, [ ]k+1,c1...cm−2 ⟩. On the coset A(m−1) in generalized Post’s Coset Theorem construct (k+ 1)-ary group ⟨A(m−1) , [ ]k+1⟩. Proved isomorphism of constructed (k + 1)-ary groups. This isomorphism allows us to prove another criterion m-semi-commutativity n-ary group: nary group ⟨A, [ ]⟩ is m-semi-abelian if and only if for some c1, . . . , cm−2 ∈ A (k + 1)-ary group ⟨A, [ ]k+1,c1...cm−2 ⟩ is abelian.