Algebra and Logic最新文献

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.

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.

For a 5-dimensional 2-step Carnot group G_3,2 with a codimension 2 horizontal distribution, we prove that any two points u, v ∈ G_3,2 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.

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.

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 (Zleft(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.

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 ²A_n(q) and ²D_n(q).

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).

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 C_G(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 C_G(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 𝔈 (g) such that for every x ∈ G all sufficiently long commutators [...[[x, g], g], . . . , g] belong to 𝔈 (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}.