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Том 57, № 1 (2018)
- Год: 2018
- Статей: 8
- URL: https://journals.rcsi.science/0002-5232/issue/view/14547
Article
1-8
Maximal and Submaximal ????-Subgroups
Аннотация
Let ???? be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup H of a finite group G a submaximal ????-subgroup if there exists an isomorphic embedding ϕ : G ↪ G* of G into some finite group G* under which Gϕ is subnormal in G* and Hϕ = K ∩ Gϕ for some maximal ????-subgroup K of G*. In the case where ???? coincides with the class of all π-groups for some set π of prime numbers, submaximal ????-subgroups are called submaximal π-subgroups. In his talk at the well-known conference on finite groups in Santa Cruz in 1979, Wielandt emphasized the importance of studying submaximal π-subgroups, listed (without proof) certain of their properties, and formulated a number of open questions regarding these subgroups. Here we prove properties of maximal and submaximal ????- and π-subgroups and discuss some open questions both Wielandt’s and new ones. One of such questions due to Wielandt reads as follows: Is it always the case that all submaximal ????-subgroups are conjugate in a finite group G in which all maximal ????-subgroups are conjugate?
9-28
Divisible Rigid Groups. II. Stability, Saturation, and Elementary Submodels
Аннотация
A group G is said to be rigid if it contains a normal series 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 divisible if elements of the quotient Gi/Gi+1 are divisible by nonzero elements of the ring ℤ[G/Gi]. Every rigid group is embedded in a divisible one. Previously, it was stated that the theory ????m of divisible m-rigid groups is complete. Here, it is proved that this theory is ω-stable. Furthermore, we describe saturated models, study elementary submodels of an arbitrary model, and find a representation for a countable saturated model in the form of a limit group in the Fraïssé system of all finitely generated m-rigid groups. Also, it is proved that the theory ????m admits quantifier elimination down to a Boolean combination of ∀∃-formulas.
29-38
Rationality of Verbal Subsets in Solvable Groups
Аннотация
A verbal subset of a group G is a set w[G] of all values of a group word w in this group. We consider the question whether verbal subsets of solvable groups are rational in the sense of formal language theory. It is proved that every verbal subset w[N] of a finitely generated nilpotent group N with respect to a word w with positive exponent is rational. Also we point out examples of verbal subsets of finitely generated metabelian groups that are not rational.
39-48
Separability of Schur Rings over Abelian p-Groups
Аннотация
A Schur ring (an S-ring) is said to be separable if each of its algebraic isomorphisms is induced by an isomorphism. Let Cn be the cyclic group of order n. It is proved that all S-rings over groups \( D={C}_p\times {C}_{p^k} \), where p ∈ {2, 3} and k ≥ 1, are separable with respect to a class of S-rings over Abelian groups. From this statement, we deduce that a given Cayley graph over D and a given Cayley graph over an arbitrary Abelian group can be checked for isomorphism in polynomial time with respect to |D|.
49-68
Centralizer Dimensions of Partially Commutative Metabelian Groups
Аннотация
We establish an upper bound for the centralizer dimension of a partially commutative metabelian group that depends linearly on the number of vertices in a defining graph. It is proved that centralizer dimensions of 2-generated metabelian groups are not bounded above. The exact value of the centralizer dimension is computed for a partially commutative metabelian group defined by a cycle.
69-80
Periodic Groups Saturated with Finite Simple Groups of Lie Type of Rank 1
Аннотация
A group G is saturated with groups from a set ℜ of groups if every finite subgroup of G is contained in a subgroup of G that is isomorphic to some group in ℜ. Previously [Kourovka Notebook, Quest. 14.101], the question was posed whether a periodic group saturated with finite simple groups of Lie type whose ranks are bounded in totality is itself a simple group of Lie type. A partial answer to this question is given for groups of Lie type of rank 1. We prove the following: Let a periodic group G be saturated with finite simple groups of Lie type of rank 1. Then G is isomorphic to a simple group of Lie type of rank 1 over a suitable locally finite field.
81-86
Sessions of the Seminar “Algebra i Logika”
87-87