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Abstract: An axiomatizability criterion is found for the class of subdirectly irreducible S-acts over a commutative monoid. As a corollary, a number of properties are presented which a commutative monoid should satisfy provided that the class of subdirectly irreducible acts over it is axiomatizable. The question about a complete description of monoids over which the class of subdirectly irreducible acts is axiomatizable remains open even for the case of a commutative monoid. PubDate: 2024-02-16

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Abstract: Inference rules are examined which are admissible immediately in all residually finite extensions of S4 possessing the weak cocover property. An explicit basis is found for such WCP-globally admissible rules. In case of tabular logics, the basis is finite, and for residually finite extensions, the independency of an explicit basis is proved. PubDate: 2024-02-15

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Abstract: The notion of an exponential R-group, where R is an arbitrary associative ring with unity, was introduced by R. Lyndon. Myasnikov and Remeslennikov refined the notion of an R-group by introducing an additional axiom. In particular, the new concept of an exponential MR-group (R-ring) is a direct generalization of the concept of an R-module to the case of noncommutative groups. We come up with the notions of a variety of MR-groups and of tensor completions of groups in varieties. Abelian varieties of MR-groups are described, and various definitions of nilpotency in this category are compared. It turns out that the completion of a 2-step nilpotent MR-group is 2-step nilpotent. PubDate: 2024-02-15

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Abstract: For a 5-dimensional 2-step Carnot group G3,2 with a codimension 2 horizontal distribution, we prove that any two points u, v ∈ G3,2 can be joined on it by a horizontal broken line consisting of at most three segments. A multi-dimensional generalization of this result is given. PubDate: 2024-02-15

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Abstract: An S-ring (Schur ring) is said to be central if it is contained in the center of a group ring. We introduce the notion of a generalized Schur group, i.e., a finite group such that all central S-rings over this group are Schurian. It generalizes the notion of a Schur group in a natural way, and for Abelian groups, the two notions are equivalent. We prove basic properties and present infinite families of non-Abelian generalized Schur groups. PubDate: 2024-02-14

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Abstract: We prove that if \(\mathcal{A}\) = (A,⋅) is a group computable in polynomial time (P-computable), then there exists a P-computable group \(\mathcal{B}\) = (B,∙) ≅ \(\mathcal{A},\) in which the operation x−1 is also P-computable. On the other hand, we show that if the center \(Z\left(\mathcal{A}\right)\) of a group A contains an element of infinite order, then under some additional assumptions, there exists a P-computable group \({\mathcal{B}}{\prime}=\left({B}{\prime},\cdot \right)\cong \mathcal{A}\) in which the operation x−1 is not primitive recursive. Also the following general fact in the theory of P-computable structures is stated: if \(\mathcal{A}\) is a P-computable structure and E ⊆ A2 is a P-computable congruence on \(\mathcal{A},\) then the quotient structure \(\mathcal{A}/E\) is isomorphic to a P-computable structure. PubDate: 2024-02-13

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Abstract: A group G is called a Shunkov group (a conjugate biprimitive finite group) if, for any of its finite subgroups H in the factor group NG(H)/H, every two conjugate elements of prime order generate a finite subgroup. We say that a group is saturated with groups from the set ð” if any finite subgroup of the given group is contained in its subgroup isomorphic to some group in ð”. We show that a Shunkov group G which is saturated with groups from the set ð” possessing specific properties, and contains an involution z with the property that the centralizer CG(z) has only finitely many elements of finite order will have a periodic part isomorphic to one of the groups in ð”. In particular, a Shunkov group G that is saturated with finite almost simple groups and contains an involution z with the property that the centralizer CG(z) has only finitely many elements of finite order will have a periodic part isomorphic to a finite almost simple group. PubDate: 2023-12-28

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Abstract: For a finite group G, the spectrum is the set ω(G) of element orders of the group G. The spectrum of G is closed under divisibility and is therefore uniquely determined by the set μ(G) consisting of elements of ω(G) that are maximal with respect to divisibility. We prove that a finite group isospectral to Aut(J2) is unsolvable. PubDate: 2023-12-28

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Abstract: It is proved that a singular superalgebra with a 2-dimensional even part is isomorphic to a superalgebra B2 3(φ, ξ, ψ). In particular, there do not exist infinite-dimensional simple singular superalgebras with a 2-dimensional even part. It is proved that if a singular superalgebra contains an odd left annihilator, then it contains a nondegenerate switch. Lastly, it is established that for any number N ≥ 5, except the numbers 6, 7, 8, 11, there exist singular superalgebras with a switch of dimension N. For the numbers N = 6, 7, 8, 11, there do not exist singular N -dimensional superalgebras with a switch. PubDate: 2023-11-08 DOI: 10.1007/s10469-023-09716-z

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Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

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Abstract: Let G be a finite group of Lie type, and T some maximal torus of the group G. We bring to a close the study of the question of whether there exists a complement for a torus T in its algebraic normalizer N (G, T). It is proved that any maximal torus of a group G ∈ {G2(q), 2G2(q), 3D4(q)} has a complement in its algebraic normalizer. Also we consider the remaining twisted classical groups 2An(q) and 2Dn(q). PubDate: 2023-03-01

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Abstract: A group G is said to be m-rigid if it contains a normal series of the form G = G1 > G2 > . . . > Gm > Gm+1 = 1, whose quotients Gi/Gi+1 are Abelian and, treated as (right) ℤ[G/Gi]-modules, are torsion-free. A rigid group G is said to be divisible if elements of the quotient ρi(G)/ρi+1(G) are divisible by nonzero elements of the ring ℤ[G/ρi(G)]. Previously, it was proved that the theory of divisible m-rigid groups is complete and ω-stable. In the present paper, we give an algebraic description of elements and types that are generic over a divisible m-rigid group G. PubDate: 2023-03-01

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Abstract: We give a description of finite 4-primary groups with disconnected Gruenberg–Kegel graph containing a triangle. As a corollary, finite groups whose Gruenberg–Kegel graph coincides with the Gruenberg–Kegel graph of 3D4(2) are exemplified, which generalizes V. D. Mazurov’ description of finite groups isospectral to the group 3D4(2). PubDate: 2023-03-01

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Abstract: There is a well-known factorization of the number 22m + 1, with m odd, related to the orders of tori of simple Suzuki groups: 22m +1 is a product of a = 2m + 2(m+1)/2 +1 and b = 2m − 2(m+1)/2 + 1. By the Bang–Zsigmondy theorem, there is a primitive prime divisor of 24m − 1, that is, a prime r that divides 24m − 1 and does not divide 2i − 1 for any 1 ≤ i < 4m. It is easy to see that r divides 22m + 1, and so it divides one of the numbers a and b. It is proved that for every m > 5, each of a, b is divisible by some primitive prime divisor of 24m − 1. Similar results are obtained for primitive prime divisors related to the simple Ree groups. As an application, we find the independence and 2-independence numbers of the prime graphs of almost simple Suzuki–Ree groups. PubDate: 2023-03-01

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Abstract: Suppose that a finite group G admits a soluble group of coprime automorphisms A. We prove that if, for some positive integer m, every element of the centralizer CG(A) has a left Engel sink of cardinality at most m (or a right Engel sink of cardinality at most m), then G has a subgroup of ( A ,m)-bounded index which has Fitting height at most 2α(A) + 2, where α(A) is the composition length of A. We also prove that if, for some positive integer r, every element of the centralizer CG(A) has a left Engel sink of rank at most r (or a right Engel sink of rank at most r), then G has a subgroup of ( A , r)-bounded index which has Fitting height at most 4α(A) + 4α(A) + 3. Here, a left Engel sink of an element g of a group G is a set ð”ˆ (g) such that for every x ∈ G all sufficiently long commutators [...[[x, g], g], . . . , g] belong to ð”ˆ (g). (Thus, g is a left Engel element precisely when we can choose (g) = {1}.) A right Engel sink of an element g of a group G is a set ℜ(g) such that for every x ∈ G all sufficiently long commutators [...[[g, x], x], . . . , x] belong to ℜ(g). Thus, g is a right Engel element precisely when we can choose ℜ(g) = {1}. PubDate: 2023-03-01

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Abstract: Let π be a proper subset of the set of all prime numbers. Denote by r the least prime number not in π, and put m = r, if r = 2, 3, and m = r − 1 if r ≥ 5. We look at the conjecture that a conjugacy class D in a finite group G generates a π-subgroup in G (or, equivalently, is contained in the π-radical) iff any m elements from D generate a π-group. Previously, this conjecture was confirmed for finite groups whose every non-Abelian composition factor is isomorphic to a sporadic, alternating, linear or unitary simple group. Now it is confirmed for groups the list of composition factors of which is added up by exceptional groups of Lie type 2B2(q), 2G2(q), G2(q), and 3D4(q). PubDate: 2023-03-01