Vol 58, No 4 (2019)

Let G be a group and S ⊆ G a subset such that S = S⁻¹, where S⁻¹ = {s⁻¹ | s ∈ S}. Then a Cayley graph Cay(G, S) is an undirected graph Γ with vertex set V (Γ) = G and edge set E(Γ) = {(g, gs) | g ∈ G, s ∈ S}. For a normal subset S of a finite group G such that s ∈ S ⇒ s^k ∈ S for every k ∈ ℤ which is coprime to the order of s, we prove that all eigenvalues of the adjacency matrix of Cay(G, S) are integers. Using this fact, we give affirmative answers to Questions 19.50(a) and 19.50(b) in the Kourovka Notebook.

It is proved that the universal equivalence of general or special linear groups of orders greater than 2 over local commutative rings with 1/2 is equivalent to the coincidence of orders of groups and universal equivalence of respective rings.

Let L(M) be a class of all groups G in which the normal closure of any element belongs to M; qM is a quasivariety generated by a class M. We consider a quasivariety qH₂ generated by a relatively free group in a class of nilpotent groups of class at most 2 with commutator subgroup of exponent 2. It is proved that the Levi class L(qH₂) generated by the quasivariety qH₂ is contained in the variety of nilpotent groups of class at most 3.

Let ℳ be a structure of a signature Σ. For any ordered tuple \( \overline{a}=\left({a}_1,\dots, {a}_{\mathrm{n}}\right) \) of elements of ℳ, \( {\mathrm{tp}}^{\mathcal{M}}\left(\overline{a}\right) \) denotes the set of formulas θ(x₁, …, x_n) of a first-order language over Σ with free variables x₁, . . . , x_n such that \( \mathcal{M}\left|=\theta \left({a}_1,\dots, {a}_n\right)\right. \). A structure ℳ is said to be strongly ω-homogeneous if, for any finite ordered tuples \( \overline{a} \) and \( \overline{b} \) of elements of ℳ, the coincidence of \( {\mathrm{tp}}^{\mathcal{M}}\left(\overline{a}\right) \) and \( {\mathrm{tp}}^{\mathrm{M}}\left(\overline{b}\right) \) implies that these tuples are mapped into each other (componentwise) by some automorphism of the structure ℳ. A structure ℳ is said to be prime in its theory if it is elementarily embedded in every structure of the theory Th (ℳ). It is proved that the integral group rings of finitely generated relatively free orderable groups are prime in their theories, and that this property is shared by the following finitely generated countable structures: free nilpotent associative rings and algebras, free nilpotent rings and Lie algebras. It is also shown that finitely generated non-Abelian free nilpotent associative algebras and finitely generated non-Abelian free nilpotent Lie algebras over uncountable fields are strongly ω-homogeneous.