Endomorphisms of semicyclic n-groups

Abstract

are isomorphic to (𝑛, 2)-nearrings of endomorphisms of certain semiabelian 𝑛-groups. Such almost (𝑛, 2)-nearrings are found for semicyclic 𝑛-groups. On the additive group of integers 𝑍 we construct an abelian 𝑛-group ⟨𝑍, 𝑓1⟩ with 𝑛- ary operation 𝑓1(𝑧1, . . . , 𝑧𝑛) = 𝑧1 + . . . + 𝑧𝑛 + 𝑙, where 𝑙 is any integer. For a nonidentical automorphism 𝜙(𝑧) = −𝑧 on 𝑍, we can specify semiabelian 𝑛-group ⟨𝑍, 𝑓2⟩ for 𝑛 = 2𝑘 + 1, 𝑘 ∈ 𝑁, with the 𝑛-ary operation 𝑓2(𝑧1, . . . , 𝑧𝑛) = 𝑧1 − 𝑧2 + . . . + 𝑧2𝑘−1 − 𝑧2𝑘 + 𝑧2𝑘+1. Any infinite semicyclic 𝑛-group is isomorphic to either the 𝑛-group ⟨𝑍, 𝑓1⟩, where 0 ≤ 𝑙 ≤ [𝑛−1 2 ], or the 𝑛-group ⟨𝑍, 𝑓2⟩ for odd 𝑛. In the first case we will say that such 𝑛-group has type (∞, 1, 𝑙), and in the second case, it has type (∞,−1, 0). In 𝑍 we select the set 𝑃 = {𝑚|𝑚𝑙 ≡ 𝑙 (mod 𝑛 − 1)} and define an 𝑛-ary operation ℎ by the rule ℎ(𝑚1, . . . ,𝑚𝑛) = 𝑚1 + . . . + 𝑚𝑛 on this set. Then the algebra ⟨𝑃, ℎ, ·⟩, where · is the multiplication of integers, is a (𝑛, 2)-ring. It is proved that ⟨𝑃, ℎ, ·⟩ is isomorphic to (𝑛, 2)-ring of endomorphisms of semicyclic 𝑛-group of type (∞, 1, 𝑙). In the 𝑛-group ⟨𝑍 × 𝑍, ℎ⟩ = ⟨𝑍, 𝑓2⟩ × ⟨𝑍, 𝑓2⟩ we define the binary operation ◇ by the rule (𝑚1, 𝑢1) ◇ (𝑚2, 𝑢2) = (𝑚1𝑚2,𝑚1𝑢2 +𝑢1). Then ⟨𝑍 ×𝑍, ℎ, ◇⟩ is an (𝑛, 2)-nearringsg. It is proved that ⟨𝑍 ×𝑍, ℎ, ◇⟩ is isomorphic to (𝑛, 2)-nearrings of endomorphisms of a semicyclic 𝑛-group of type (∞,−1, 0). It is proved that (𝑛, 2)-ring ⟨𝑍, 𝑓, *⟩, where 𝑓(𝑧1, . . . , 𝑧𝑛) = 𝑧1 + . . . + 𝑧𝑛 + 1 and 𝑧1 * 𝑧2 = 𝑧1𝑧2(𝑛 − 1) + 𝑧1 + 𝑧2, is isomorphic to (𝑛, 2)-rings of endomorphisms of infinite cyclic 𝑛-group. On additive group of the ring of residue classes of 𝑍𝑘 we define 𝑛-group ⟨𝑍𝑘, 𝑓3⟩, where the 𝑛- ary operation 𝑓3 operates according to the rule 𝑓3(𝑧1, . . . , 𝑧𝑛) = 𝑧1+𝑚𝑧2+. . .+𝑚𝑛−2𝑧𝑛−1+𝑧𝑛+𝑙, 1 ≤ 𝑚 < 𝑘 and 𝑚 is relatively prime to 𝑘. In addition, 𝑚 satisfies the congruence 𝑙𝑚 ≡ 𝑙 (mod 𝑘) and multiplicative order of 𝑚 modulo 𝑘 divides 𝑛 − 1. Any finite semicyclic 𝑛-group of order 𝑘 is isomorphic to 𝑛-group ⟨𝑍𝑘, 𝑓3⟩, where 𝑙 | gcd(𝑛 − 1, 𝑘) for 𝑚 = 1 and 𝑙 | gcd(𝑚𝑛−1−1 𝑚−1 , 𝑘) for 𝑚 = 1. We will say that such 𝑛-group has type (𝑘, 𝑚, 𝑙). In the 𝑛-group ⟨𝑃, ℎ⟩ = ⟨𝑍𝑘, 𝑓3⟩ × ⟨𝑍𝑙, 𝑓4⟩, 𝑓4(𝑧1, . . . , 𝑧𝑛) = 𝑧1 + 𝑟𝑧2 + . . . + 𝑟𝑛−2𝑧𝑛−1 + 𝑧𝑛, where 𝑟 is the remainder of dividing 𝑚 by 𝑙, we define the binary operation ◇ by the rule $$(𝑢1, 𝑣1) ◇ (𝑢2, 𝑣2) = (𝑢2𝑠1 + 𝑢1, 𝑣2𝑠1 + 𝑣1)$$ where 𝑠1 ∈ 𝑍𝑘, (𝑠1 − 1 = 𝑠0 + 𝑣1)/𝑘 𝑙 , and 𝑠0 is solution of congruence 𝑥 ≡ (𝑛−1)𝑢1 𝑙 (mod 𝑘/𝑙 ) for 𝑚 = 1 and 𝑥 ≡ 𝑚𝑛−1−1 𝑚−1 𝑢1 𝑙 (mod 𝑘/𝑙 ) for 𝑚 = 1. It is proved that the algebra ⟨𝑃, ℎ, ◇⟩ is (𝑛, 2)-ring for 𝑚 = 1 and (𝑛, 2)-nearring for 𝑚 = 1, which is isomorphic to (𝑛, 2)-ring of endomorphisms of abelian semicyclic 𝑛-group of type (𝑘, 1, 𝑙) with 𝑚 = 1 and (𝑛, 2)-nearring of endomorphisms of semicyclic 𝑛-groups of type (𝑘, 𝑚, 𝑙) for 𝑚 = 1. It is proved that (𝑛, 2)-ring ⟨𝑍𝑘, 𝑓, *⟩, where 𝑓(𝑧1, 𝑙𝑑𝑜𝑡𝑠, 𝑧𝑛) = 𝑧1 + 𝑙𝑑𝑜𝑡𝑠 + 𝑧𝑛 + 1 and 𝑢1 *𝑢2 = 𝑢1 𝑐𝑑𝑜𝑡𝑢2 𝑐𝑑𝑜𝑡(𝑛−1)+𝑢1+𝑢2, is isomorphic to (𝑛, 2)-ring of endomorphisms of finite cyclic 𝑛-group of order 𝑘.