Authors:A. A. Buturlakin Pages: 1 - 8 Abstract: We give a description of spectra of finite simple and universal groups of Lie type E8. PubDate: 2018-03-01 DOI: 10.1007/s10469-018-9474-9 Issue No:Vol. 57, No. 1 (2018)

Authors:W. Guo; D. O. Revin Pages: 9 - 28 Abstract: Let ð”› 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 ð”›-subgroup if there exists an isomorphic embedding ϕ : G ↪ G * of G into some finite group G * under which G ϕ is subnormal in G * and H ϕ = K ∩ G ϕ for some maximal ð”›-subgroup K of G * . In the case where ð”› coincides with the class of all π-groups for some set π of prime numbers, submaximal ð”›-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 ð”›- 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 ð”›-subgroups are conjugate in a finite group G in which all maximal ð”›-subgroups are conjugate' PubDate: 2018-03-01 DOI: 10.1007/s10469-018-9475-8 Issue No:Vol. 57, No. 1 (2018)

Authors:A. G. Myasnikov; N. S. Romanovskii Pages: 29 - 38 Abstract: A group G is said to be rigid if it contains a normal series G = G1 > G2 > . . . > G m > Gm+1 = 1, whose quotients G i /Gi+1 are Abelian and, treated as right ℤ[G/G i ]- modules, are torsion-free. A rigid group G is divisible if elements of the quotient G i /Gi+1 are divisible by nonzero elements of the ring ℤ[G/G i ]. Every rigid group is embedded in a divisible one. Previously, it was stated that the theory ð”— 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 ð”— m admits quantifier elimination down to a Boolean combination of ∀∃-formulas. PubDate: 2018-03-01 DOI: 10.1007/s10469-018-9476-7 Issue No:Vol. 57, No. 1 (2018)

Authors:V. A. Roman’kov Pages: 39 - 48 Abstract: 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. PubDate: 2018-03-01 DOI: 10.1007/s10469-018-9477-6 Issue No:Vol. 57, No. 1 (2018)

Authors:G. K. Ryabov Pages: 49 - 68 Abstract: A Schur ring (an S-ring) is said to be separable if each of its algebraic isomorphisms is induced by an isomorphism. Let C n 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 . PubDate: 2018-03-01 DOI: 10.1007/s10469-018-9478-5 Issue No:Vol. 57, No. 1 (2018)

Authors:E. I. Timoshenko Pages: 69 - 80 Abstract: 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. PubDate: 2018-03-01 DOI: 10.1007/s10469-018-9479-4 Issue No:Vol. 57, No. 1 (2018)

Authors:A. A. Shlepkin Pages: 81 - 86 Abstract: 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. PubDate: 2018-03-01 DOI: 10.1007/s10469-018-9480-y Issue No:Vol. 57, No. 1 (2018)

Authors:N. A. Bazhenov; M. I. Marchuk Abstract: We look at the concept of algorithmic complexity of isomorphisms between computable copies of Boolean algebras. Degrees of autostability are found for all prime Boolean algebras. It is shown that for any ordinals α and β with the condition 0 ≤ α ≤ β ≤ ω, there is a decidable model for which 0(α) is a degree of autostability relative to strong constructivizations, while 0(β) is a degree of autostability. It is proved that for any nonzero ordinal β ≤ ω, there is a decidable model for which there is no degree of autostability relative to strong constructivizations, while 0(β) is a degree of autostability. PubDate: 2018-07-25 DOI: 10.1007/s10469-018-9483-8

Authors:A. A. Makhnev; D. V. Paduchikh; L. Yu. Tsiovkina Abstract: We complete the classification of edge-symmetric distance-regular coverings of complete graphs with r /∉ {2, k, (k − 1)/μ} for the case of the almost simple action of an automorphism group of a graph on a set of its antipodal classes; here r is the order of an antipodal class. PubDate: 2018-07-20 DOI: 10.1007/s10469-018-9486-5

Authors:I. B. Gorshkov; N. V. Maslova Abstract: It is shown that the Gruenberg–Kegel graph of a finite almost simple group is equal to the Gruenberg–Kegel graph of some finite solvable group iff it does not contain 3-cocliques. Furthermore, we obtain a description of finite almost simple groups whose Gruenberg–Kegel graphs contain no 3-cocliques. PubDate: 2018-07-20 DOI: 10.1007/s10469-018-9484-7

Authors:A. I. Sozutov; O. V. Kravtsova Abstract: We study KT-fields and sharply triply transitive groups with a finite or perfect involution stabilizing at least one point. PubDate: 2018-07-20 DOI: 10.1007/s10469-018-9487-4

Authors:M. G. Amaglobeli Abstract: The notion of an exponential R-group, where R is an arbitrary associative ring with unity, was introduced by R. Lyndon. A. G. Myasnikov and V. N. Remeslennikov refined this notion by adding an extra axiom. In particular, the new notion of an exponential MR-group is an immediate generalization of the notion of an R-module to the case of noncommutative groups. Basic concepts in the theory of exponential MR-groups are presented, and we propose a particular method for constructing tensor completion—the key construction in the category of MR-groups. As a consequence, free MR-groups and free MR-products are described using the language of group constructions. PubDate: 2018-07-20 DOI: 10.1007/s10469-018-9482-9

Authors:A. N. Zubkov Abstract: Some standard theorems on Noetherian schemes are generalized to the case of Noetherian superschemes. PubDate: 2018-07-20 DOI: 10.1007/s10469-018-9485-6

Authors:D. O. Ptakhov Abstract: Polygons with a (P, 1)-stable theory are considered. A criterion of being (P, 1)-stable for a polygon is established. As a consequence of the main criterion we prove that a polygon SS, where S is a group, is (P, 1)-stable if and only if S is a finite group. It is shown that the class of all polygons with monoid S is (P, 1)-stable only if S is a one-element monoid. (P, 1)-stability criteria are presented for polygons over right and left zero monoids. PubDate: 2018-02-24 DOI: 10.1007/s10469-018-9469-6

Authors:E. P. Vdovin; M. N. Nesterov; D. O. Revin Abstract: It is known that for any set π of prime numbers, the following assertions are equivalent: (1) in any finite group, π-Hall subgroups are conjugate; (2) in any finite group, π-Hall subgroups are pronormal. It is proved that (1) and (2) are equivalent also to the following: (3) in any finite group, π-Hall subgroups are pronormal in their normal closure. Previously [10, Quest. 18.32], the question was posed whether it is true that in a finite group, π-Hall subgroups are always pronormal in their normal closure. Recently, M. N. Nesterov [7] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set π. The fact that there exist examples of finite sets π and finite groups G such that G contains more than one conjugacy class of π-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for π is unessential for (1), (2), and (3) to be equivalent. PubDate: 2018-02-24 DOI: 10.1007/s10469-018-9467-8