Homomorphisms from infinite semilcyclic n-groups to a semiabelian n-group

Abstract

which are isomorphic to (𝑛)-groups of homomorphisms from certain 𝑛-groups to a semiabelian 𝑛-group. Such 𝑛-groups are found for infinite semicyclic 𝑛-groups. It is known that the set 𝐻𝑜𝑚(𝐺,𝐶) of all homomorphisms from 𝑛-groups ⟨𝐺, 𝑓1⟩ to a semiabelian (abelian) 𝑛-group ⟨𝐶, 𝑓2⟩ with 𝑛-ary operation 𝑔 given by the rule 𝑔(𝜙1, . . . , 𝜙𝑛)(𝑥) = 𝑓2(𝜙1(𝑥), . . . , 𝜙𝑛(𝑥)), 𝑥 ∈ 𝐺, forms a semiabelian (abelian) 𝑛-group. It is proved that the isomorphisms 𝜓1 of 𝑛-groups ⟨𝐺, 𝑓1⟩ and ⟨𝐺′, 𝑓′1 ⟩ and 𝑝𝑠𝑖2 of semiabelian 𝑛-groups ⟨𝐶, 𝑓2⟩ and ⟨𝐶′, 𝑓′2⟩ induce an isomorphism 𝜏 of 𝑛-groups of homomorphisms ⟨𝐻𝑜𝑚(𝐺,𝐶), 𝑔⟩ and ⟨𝐻𝑜𝑚(𝐺′,𝐶′), 𝑔′⟩, which acts according to the rule 𝜏 : 𝛼 → 𝜓2 ∘ 𝛼 ∘ 𝜓−1 1 .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 studying the 𝑛-groups of homomorphisms ⟨𝐻𝑜𝑚(𝑍,𝐶), 𝑔⟩ from an infinite abelian semicyclic 𝑛-group ⟨𝑍, 𝑓1⟩ (0 ≤ 𝑙 ≤ 𝑛−1 2 ) to a semiabelian 𝑛-group ⟨𝐶, 𝑓2⟩ we construct on the 𝑛-group ⟨𝐶, 𝑓2⟩ an abelian group 𝐶 with the addition operation 𝑎 + 𝑏 = 𝑓2(𝑎, (𝑛−3) 𝑐 , ¯𝑐, 𝑏), in which there is an element 𝑑2 = 𝑓2( (𝑛) 𝑐 ) and an automorphism 𝜙2(𝑥) = 𝑓2(𝑐, 𝑥, (𝑛−3) 𝑐 , ¯𝑐). Choose a set 𝑃1 of such ordered pairs (𝑎, 𝑢) of elements from 𝐶 that satisfy the equality 𝑙𝑎 = 𝑑2 + ∼𝜙2(𝑢), where ∼𝜙2(𝑥) = 𝑥 + 𝜙2(𝑥) + . . . + 𝜙𝑛−2 2 (𝑥), 𝑥 ∈ 𝐶 is an endomorphism of the group 𝐶, and for the first component of these pairs the equality is true 𝜙2(𝑎) = 𝑎. On this set, we define a 𝑛-ary operation ℎ1 by the rule ℎ1((𝑎1, 𝑢1), . . . , (𝑎𝑛, 𝑢𝑛)) = (𝑎1 + . . . + 𝑎𝑛, 𝑓2(𝑢1, . . . , 𝑢𝑛)). It is proved that ⟨𝑃1, ℎ1⟩ is a semiabelian 𝑛-group, which is isomorphic to the 𝑛-group of homomorphisms from an infinite abelian semicyclic 𝑛-group ⟨𝑍, 𝑓1⟩ (0 ≤ 𝑙 ≤ 𝑛−1 2 ) to an 𝑛-group ⟨𝐶, 𝑓2⟩. The consequence of this isomorphism is an isomorphism of 𝑛-groups of ⟨𝑃1, ℎ1⟩ and 𝑛-groups of homomorphisms from an infinite abelian semicyclic 𝑛-group of type (∞, 1, 𝑙) to a semiabelian 𝑛-group ⟨𝐶, 𝑓2⟩. When studying the 𝑛-group of homomorphisms ⟨𝐻𝑜𝑚(𝑍,𝐶), 𝑔⟩ from the infinite semicyclic 𝑛-group ⟨𝑍, 𝑓′1 ⟩ to the semiabelian 𝑛-group ⟨𝐶, 𝑓2⟩ in the abelian group 𝐶 choose the subgroup 𝐻 = {𝑎 ∈ 𝐶 | 𝜙2(𝑎) = −𝑎}. On 𝐻 we define a semiabelian 𝑛-group ⟨𝐻, ℎ⟩, where ℎ acts according to the rule ℎ(𝑎1, 𝑎2, . . . , 𝑎𝑛−1, 𝑎𝑛) = 𝑎1 +𝜙2(𝑎2)+. . .+𝜙𝑛−2 2 (𝑎𝑛−1)+𝑎𝑛. Then in the 𝑛-group ⟨𝐶, 𝑓2⟩ we select the subgroup ⟨𝑇, 𝑓2⟩ of all dempotents, if 𝑇 = ∅. It is proved that foran odd number 𝑛 > 1 a direct product of semiabelian 𝑛-groups ⟨𝐻, ℎ⟩ × ⟨𝑇, 𝑓2⟩ is isomorphic to 𝑛-group of homomorphisms from infinite semicyclic 𝑛-groups of ⟨𝑍, 𝑓′ 1⟩ to a semiabelian 𝑛-group ⟨𝐶, 𝑓2⟩ with a non empty set of idempotents 𝑇. The consequence of this isomorphism is the isomorphism of the 𝑛-group ⟨𝐻, ℎ⟩ × ⟨𝑇, 𝑓2⟩ and 𝑛-groups of homomorphisms from an infinite semicyclic 𝑛-group of type (∞,−1, 0) to the semiabelian 𝑛-group ⟨𝐶, 𝑓2⟩. Similar facts were obtained when studying the 𝑛-group of homomorphisms ⟨𝐻𝑜𝑚(𝑍,𝐶), 𝑔⟩ from 𝑛-groups ⟨𝑍, 𝑓1⟩ and ⟨𝑍, 𝑓′ 1⟩ to an abelian 𝑛-group ⟨𝐶, 𝑓2⟩