Authors:W. Guo; D. O. Revin Pages: 169 - 181 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 isomorpic embedding ϕ: G ↪ G* of the group G into some finite group G* under which Gϕ is subnormal in G* and Hϕ = K ∩Gϕ for some maximal ð”›-subgroup K of G*. We discuss the following question formulated by Wielandt: 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? This question strengthens Wielandt’s known problem of closedness for the class of -groups under extensions, which was solved some time ago. We prove that it is sufficient to answer the question mentioned for the case where G is a simple group. PubDate: 2018-07-01 DOI: 10.1007/s10469-018-9490-9 Issue No:Vol. 57, No. 3 (2018)

Authors:D. K. Kabylzhanova Pages: 182 - 185 Abstract: We consider positive preorders, i.e., computably enumerable equivalences, endowed with the structure of a partial order between equivalence classes. On positive preorders, a computable reducibility relation and the corresponding notion of degree of a positive preorder are introduced in the natural way. It is proved that the degree of any positive preorder contains either exactly one computable isomorphism class or an infinite set of computable isomorphism classes. PubDate: 2018-07-01 DOI: 10.1007/s10469-018-9491-8 Issue No:Vol. 57, No. 3 (2018)

Authors:S. S. Korobkov Pages: 186 - 200 Abstract: Associative rings R and R′ are said to be lattice-isomorphic if their subring lattices L(R) and L(R′) are isomorphic. An isomorphism of the lattice L(R) onto the lattice L(R′) is called a projection (or a lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. We study lattice isomorphisms of finite commutative rings with identity. The objective is to specify sufficient conditions subject to which rings under lattice homomorphisms preserve the following properties: to be a commutative ring, to be a ring with identity, to be decomposable into a direct sum of ideals. We look into the question about the projective image of the Jacobson radical of a ring. In the first part, the previously obtained results on projections of finite commutative semiprime rings are supplemented with new information. Lattice isomorphisms of finite commutative rings decomposable into direct sums of fields and nilpotent ideals are taken up in the second part. Rings definable by their subring lattices are exemplified. Projections of finite commutative rings decomposable into direct sums of Galois rings and nilpotent ideals are considered in the third part. It is proved that the presence in a ring of a direct summand definable by its subring lattice (i.e., the Galois ring GR(pn,m), where n > 1 and m > 1) leads to strong connections between the properties of R and R′. PubDate: 2018-07-01 DOI: 10.1007/s10469-018-9492-7 Issue No:Vol. 57, No. 3 (2018)

Authors:D. V. Lytkina; V. D. Mazurov Pages: 201 - 210 Abstract: Let G be a periodic group containing an element of order 2 such that each of its finite subgroups of even order lies in a finite subgroup isomorphic to a simple symplectic group of degree 4. It is shown that G is isomorphic to a simple symplectic group S4(Q) of degree 4 over some locally finite field Q. PubDate: 2018-07-01 DOI: 10.1007/s10469-018-9493-6 Issue No:Vol. 57, No. 3 (2018)

Authors:A. T. Nurtazin Pages: 211 - 221 Abstract: Necessary and sufficient conditions are stated for an arbitrary theory to be an elementary theory for a class of its existentially closed models. Conditions are given under which some existentially closed model simultaneously realizes one maximal existential type and omits another. We also prove a theorem on a prime existentially closed model over a maximal existential type. Considerable complexity of existentially closed structures and their theories was noted by A. Macintyre. Therefore, the examples of existentially closed companions having any finite or countable number of pairwise non elementarily equivalent existentially closed models constructed here are of interest. PubDate: 2018-07-01 DOI: 10.1007/s10469-018-9494-5 Issue No:Vol. 57, No. 3 (2018)

Authors:V. L. Selivanov; M. M. Yamaleev Pages: 222 - 236 Abstract: We investigate the problem of characterizing proper levels of the fine hierarchy (up to Turing equivalence). It is known that the fine hierarchy exhausts arithmetical sets and contains as a small fragment finite levels of Ershov hierarchies (relativized to ∅n, n < ω), which are known to be proper. Our main result is finding a least new (i.e., distinct from the levels of the relativized Ershov hierarchies) proper level. We also show that not all new levels are proper. PubDate: 2018-07-01 DOI: 10.1007/s10469-018-9495-4 Issue No:Vol. 57, No. 3 (2018)

Authors:M. P. Shushpanov Pages: 237 - 247 Abstract: It is known that a modular 3-generated lattice is always finite and contains at most 28 elements. Lattices generated by three elements with certain modularity properties may no longer be modular but nevertheless remain finite. It is shown that a 3-generated lattice among generating elements of which one is seminormal and another is coseminormal is finite and contains at most 45 elements. This estimate is stated to be sharp. PubDate: 2018-07-01 DOI: 10.1007/s10469-018-9496-3 Issue No:Vol. 57, No. 3 (2018)

Authors:M. G. Amaglobeli Pages: 89 - 97 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: 2018-06-01 DOI: 10.1007/s10469-018-9482-9 Issue No:Vol. 57, No. 2 (2018)

Authors:N. A. Bazhenov; M. I. Marchuk Pages: 98 - 114 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: 2018-06-01 DOI: 10.1007/s10469-018-9483-8 Issue No:Vol. 57, No. 2 (2018)

Authors:I. B. Gorshkov; N. V. Maslova Pages: 115 - 129 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: 2018-06-01 DOI: 10.1007/s10469-018-9484-7 Issue No:Vol. 57, No. 2 (2018)

Authors:A. N. Zubkov Pages: 130 - 140 Abstract: Some standard theorems on Noetherian schemes are generalized to the case of Noetherian superschemes. PubDate: 2018-06-01 DOI: 10.1007/s10469-018-9485-6 Issue No:Vol. 57, No. 2 (2018)

Authors:A. A. Makhnev; D. V. Paduchikh; L. Yu. Tsiovkina Pages: 141 - 152 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: 2018-06-01 DOI: 10.1007/s10469-018-9486-5 Issue No:Vol. 57, No. 2 (2018)

Authors:A. I. Sozutov; O. V. Kravtsova Pages: 153 - 160 Abstract: We study KT-fields and sharply triply transitive groups with a finite or perfect involution stabilizing at least one point. PubDate: 2018-06-01 DOI: 10.1007/s10469-018-9487-4 Issue No:Vol. 57, No. 2 (2018)

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: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: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)