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Abstract: For a finite group G, we denote by N (G) the set of its conjugacy class sizes. Recently, the following question was posed: given any n ∈ ℕ and an arbitrary non-Abelian finite simple group S, is it true that G ≃ Sn if G is a group with trivial center and N (G) = N (Sn)' The answer to this question is known for all simple groups S with n = 1, and also for S ∈ {A5, A6}, where Ak denotes the alternating group of degree k, with n = 2. It is proved that the group A5 × A5 × A5 is uniquely defined by the set N (A5 × A5 × A5) in the class of finite groups with trivial center. PubDate: 2024-05-01
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Abstract: We study periodic groups saturated with finite simple symplectic groups. PubDate: 2024-05-01
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Abstract: According to G. Birkhoff, there is a categorical duality between the category of bi-algebraic distributive (0, 1)-lattices with complete (0, 1)-lattice homomorphisms as morphisms and the category of partially ordered sets with partial order-preserving maps as morphisms. We extend this classical result to the bi-algebraic lattices belonging to the variety of (0, 1)-lattices generated by the pentagon, the 5-element nonmodular lattice. Applying the extended duality, we prove that the lattice of quasivarieties contained in the variety of (0, 1)-lattices generated by the pentagon has uncountably many elements and is not distributive. This yields the following: the lattice of quasivarieties contained in a nontrivial variety of (0, 1)-lattices either is a 2-element chain or has uncountably many elements and is not distributive. PubDate: 2024-05-01
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Abstract: We generalize the constructions of Q- and G-families of quandles introduced in the paper of A. Ishii et al. in [Ill. J. Math., 57, No. 3, 817-838 (2013)], and establish how they are related to other constructions of quandles. A composition of structures of quandles defined on the same set is specified, and conditions are found under which this composition yields a quandle. It is proved that under such a multiplication we obtain a group that will be Abelian. Also a direct product of quandles is examined. PubDate: 2024-05-01
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Abstract: We study degrees and degree spectra of groups $${\mathfrak{G}}_{\mathrm{I}}$$ defined on a set of permutations of the natural numbers ω whose degrees belong to a Turing ideal I. A necessary condition and a sufficient condition are stated which specify whether an arbitrary Turing degree belongs to the degree spectrum of a group $${\mathfrak{G}}_{\mathrm{I}}$$. Nonprincipal ideals I for which the group $${\mathfrak{G}}_{\mathrm{I}}$$ has or does not have a degree are exemplified. PubDate: 2024-05-01
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Abstract: We find a bound for the length of a conjunction of some propositional formulas, for which every unsatisfiable formula contains an unsatisfiable subformula. In particular, this technique applies to formulas in conjunctive normal form with restrictions on the number of true literals within every elementary disjunction, as well as for 2-CNFs, for symmetric 3-CNFs, and for conjunctions of voting functions in three literals. A lower bound on the rank of some matrices is used in proofs. PubDate: 2024-03-01
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Abstract: A finitely generated group G, which acts on a tree so that all edge stabilizers are infinite cyclic groups and all vertex stabilizers are free rank 2 Abelian groups, is called a tubular group. Every tubular group is isomorphic to the fundamental group π1(ð’¢) of a suitable finite graph ð’¢ of groups. We prove a criterion for residual π-finiteness of tubular groups presented by trees of groups. Also we state a criterion for residual p-finiteness of tubular groups whose corresponding graph contains one edge outside a maximal subtree. PubDate: 2024-03-01
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Abstract: On a Cartan group $${\mathbb{K}}$$ equipped with a Carnot–Carathéodory metric dcc, we find the exact value of a constant in the (1, q2)-generalized triangle inequality for its Box-quasimetric. It is proved that any two points x, y ∈ $${\mathbb{K}}$$ can be joined by a horizontal k-broken line $${L}_{x,y}^{k}$$, k ≤ 6; moreover, the length of such a broken line $${L}_{x,y}^{k}$$ does not exceed the quantity Cdcc(x, y) for some constant C not depending on the choice of x, y ∈ $${\mathbb{K}}$$. The value 6 here is nearly optimal. PubDate: 2024-03-01
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Abstract: It is proved that if G is a group without elements of order 2, and the normal closure of every 2-generated subgroup of G is a nilpotent group of class at most 3, then G will be a nilpotent group of class at most 4. It is also shown that the restriction on second-order elements cannot be lifted. PubDate: 2024-03-01
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Abstract: We introduce a bipolar classification with index j for endomorphisms of an arbitrary n-groupoid with n > 1, where j = 1, 2, . . . , n. The classifications of endomorphisms constructed generalize the bipolar classification of endomorphisms of an arbitrary groupoid (i.e., a 2-groupoid) introduced previously. Using a left bipolar classification of endomorphisms of an n-groupoid (a particular case of the obtained classifications), we succeed in constructing an integral classification of endomorphisms of an arbitrary algebra (i.e., a structure without relations) with finitary operations. PubDate: 2024-03-01
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Abstract: It is proved that, for any natural n, the subalgebra generated by words of length divisible by n on generators (the Veronese n-subalgebra) in a free finitely generated alternative algebra is finitely generated. PubDate: 2024-03-01
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Abstract: Relations between some constructions based on incidence rings and group rings are considered. PubDate: 2024-03-01
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Abstract: A finite Frobenius group in which the order of complements is divisible by a prime number p is called a Φp-group. We prove the theorem stating the following. Let G be a periodic group with a finite element a of prime order p > 2 saturated with Φp-groups. Then G = F λ H is a Frobenius group with kernel F and complement H. If G contains an involution i commuting with the element a, then H = CG(i) and F is Abelian, and H = NG($$\langle a\rangle $$) otherwise. PubDate: 2024-01-01
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Abstract: If a certain condition holds for a quasivariety K then K contains continuum many subquasivarieties having a finitely partitionable ω-independent quasi-equational basis relative to K. This is true, in particular, for each almost ff-universal quasivariety K. PubDate: 2024-01-01
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Abstract: A variety of associative algebras is nonmatrix if it does not contain the algebra of 2 × 2 matrices over a given field. Nonmatrix varieties were introduced and studied by V. N. Latyshev in [Algebra and Logic, 16, No. 2, 98-122 (1977); Algebra and Logic, 16, No. 2, 122-133 (1977); Mat. Zam., 27, No. 1, 147-156 (1980)] in connection with the Specht problem. A series of equivalent characterizations of nonmatrix varieties was obtained in [Isr. J. Math., 181, No. 1, 337-348 (2011)]. In the present paper, the notion of nonmatrix variety is extended to nonassociative algebras, and their characterization from the last-mentioned paper is generalized to alternative, Jordan, and some other varieties of algebras. PubDate: 2024-01-01
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Abstract: A Levi class $$L\left(\mathcal{M}\right)$$ generated by a class $$\left(\mathcal{M}\right)$$ of groups is the class of all groups in which the normal closure of every cyclic subgroup belongs to $$\left(\mathcal{M}\right)$$. Let p be a prime and p ≠ 2, let Hp be a free group of rank 2 in the variety of nilpotent groups of class at most 2 with commutator subgroup of exponent p, and let qHp be the quasivariety generated by the group Hp. It is shown that there exists a set of quasivarieties $$\mathcal{M}$$ of cardinality continuum such that $$L\left(\mathcal{M}\right)$$ = L(qHp). Let s be a natural number, s ≥ 2. We specify a system of quasi-identities defining L(q(Hp, $${Z}_{{p}^{s}}$$)), and prove that there exists a set of quasivarieties $$\mathcal{M}$$ of cardinality continuum such that $$L\left(\mathcal{M}\right)$$ = L(q(Hp, $${Z}_{{p}^{s}}$$)), where $${Z}_{{p}^{s}}$$ is a cyclic group of order ps; q(Hp, $${Z}_{{p}^{s}}$$) is the quasivariety generated by the groups Hp and $${Z}_{{p}^{s}}.$$ PubDate: 2024-01-01
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Abstract: We will look into the following conjecture, which, if valid, would allow us to formulate an unimprovable analog of the Baer–Suzuki theorem for the π-radical of a finite group (here π is an arbitrary set of primes). For an odd prime number r, put m = r, if r = 3, and m = r - 1 if r ≥ 5. Let L be a simple non-Abelian group whose order has a prime divisor s such that s = r if r divides L and s > r otherwise. Suppose also that x is an automorphism of prime order of L. Then some m conjugates of x in the group $$\langle L,x\rangle $$ generate a subgroup of order divisible by s. The conjecture is confirmed for the case where L is a group of Lie type and x is an automorphism induced by a unipotent element. PubDate: 2024-01-01
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