We prove that every locally inner (class-preserving) endomorphism of adjoint Chevalley groups and their elementary subgroups over commutative rings is inner for the root systems A1, A2, B2 (assuming 2 is invertible in the ring), and for G2 (assuming 2 and 3 are invertible). As a consequence, these groups are Sha-rigid. The proofs are direct and do not rely on classification of automorphisms or structural results about injective endomorphisms.

In this paper, we prove that every automorphism of a Chevalley group of type F_4 over a commutative local ring without 1/2 is a composition of a ring automorphism, a graph automorphism, and a conjugation by some matrix in the adjoint representation space.

In this paper, we prove that the endomorphism rings End A and End A' of periodic infinite Abelian groups A and A' are elementarily equivalent if and only if the endomorphism rings of their p-components are elementarily equivalent for all primes p. Additionally, we show that the automorphism groups Aut A and Aut A' of periodic Abelian groups A and A' that do not have 2-components and do not contain cocyclic p-components are elementarily equivalent if and only if, for any prime p, the corresponding p-components A_p and A_p' of A and A' are equivalent in second-order logic if they are not reduced, and are equivalent in second-order logic bounded by the cardinalities of their basic subgroups if they are reduced. For such groups A and A', their automorphism groups are elementarily equivalent if and only if their endomorphism rings are elementarily equivalent, and the automorphism groups of the corresponding p-components for all primes p are elementarily equivalent.

In this paper we consider Chevalley groups over commutative rings with $1$, constructed by irreducible root systems of rank $>1$. We always suppose that for the systems $A_2, B_\ell, C_\ell, F_4, G_2$ our rings contain $1/2$ and for the system $G_2$ also $1/3$. Under these assumptions we prove that the central quotients of Chevalley groups are regularly bi-interpretable with the corresponding rings, the class of all central quotients of Chevalley groups of a given type is elementarily definable and even finitely axiomatizable. The same holds for adjoint Chevalley groups and for other Chevalley groups with some special condition of bounded generation. We also give an example of Chevalley group with infinite center, which is not bi-interpretable with the corresponding ring and is elementarily equivalent to a group that is not a Chevalley group itself.

We prove that every class-preserving endomorphism of the adjoint Chevalley group and of its elementary subgroup over a commutative ring is inner for the types A1, A2, and B2 when 2 is invertible, and for type G2 when 2 and 3 are invertible. Consequently, all these groups are Sha-rigid.

In this paper we study (logical) types and isotypical equivalence of torsion free Abelian groups. We describe all possible types of elements and standard 2-tuples of elements in these groups and classify separable torsion free Abelian groups up to isotypicity.

In this paper, a complete set of invariants that characterize isotypically equivalent Abelian periodic groups is provided. Additionally, all types of standard tuples of elements in these groups are described. As a corollary, an example of a continuum of non-isomorphic isotypically equivalent countable Abelian p-groups is given.

We prove that every locally inner endomorphism of a Chevalley group (or its elementary subgroup) over a local ring with an irreducible root system of rank >1 (with 1/2 for the systems A_2, F_4, B_l, C_l and with 1/3 for the system G_2) is inner, so that all these groups are Sha-rigid.

In this paper we prove the criterion of elementary equivalence of linear groups over graded rings with finite number of centralidempotents from the 0-component, when grading is partially included in the group language. * This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.

We prove that two Abelian $p$-groups with separable reduced parts are isotypically equivalent if and only if their divisible parts and their basic subgroups are elementarily equivalent. Also as a corollary we prove that any Abelian $p$-group with a separable reduced part is $\omega$-strongly homogeneous

In this paper, we prove a criterion of elementary equivalence of stable linear groups over fields of characteristic two.

In this paper we study the Diophantine problem in Chevalley groups Gπ(Φ,R), where Φ is a reduced irreducible root system of rank >1, R is an arbitrary commutative ring with 1.We establish a variant of double centralizer theorem for elementary unipotents xα(1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup Xα, α∈Φ, is Diophantine in G. Then we prove that the Diophantine problem in Gπ(Φ,R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.

In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank >1 over a commutative ring (with 1/2 for the systems A_2, F_4, B_l, C_l; with 1/2 and 1/3 for the system G_2) is standard, i.e., it is a composition of ring, inner, central and graph automorphisms. This result finalizes description of automorphisms of Chevalley groups. However the restrictions on invertible elements can be a topic of further considerations. We provide also some model-theoretic applications of this description.

In this paper we prove that every automorphism of a Chevalley group with the root system G_2 over a commutative ring R with 1/3, generated by all its invertible elements and the ideal 2R is a composition of ring and inner automorphisms.

In this paper, we prove a criterion of universal equivalence of symplectic linear groups over fields: two symplectic linear groups Sp2n(K) and Sp2m(M), where n,m ≥ 1 and K and M are infinite fields of characteristic not equal to 2, are universally equivalent if and only if n = m and the fields K and M are universally equivalent.

Let R be a linearly ordered commutative ring with 1 / 2 generated by its invertible elements, G 2 ( R ) be the subsemigroup in GL 2 ( R ) consisting of all matrices with nonnegative elements. In this paper, we describe endomorphisms of the given semigroup.

We prove that if G(R)=G_π (Φ ,R) (E(R)=E_π(Φ , R)) is an (elementary) Chevalley group of rank > 1 , R is a local ring (with 1/2 for the root systems A_2, B_l, C_l, F_4, G_2 and with 1/3 for G_2) , then the group G(R) (or (E(R)) is regularly bi-interpretable with the ring R. As a consequence of this theorem, we show that the class of all Chevalley groups over local rings (with the listed restrictions) is elementarily definable, i.e., if for an arbitrary group H we have H≡ G_π (Φ , R) , then there exists a ring R'≡ R such that H≅ G_π (Φ ,R') .

Victor Timofeevich was a man of very broad views, he was interested in almost everything in life, and everywhere he showed his outstanding talents and deep intelligence.In his young years, he supported many university traditions of leisure: sports, music, tourism

The prominent Soviet and Russian mathematician, outstanding teacher, Professor of the Department of Higher Algebra of the Faculty of Mechanics and Mathematics at Moscow State University Victor Nikolaevich Latyshev passed away on 13 April 2020. Latyshev was born in Moscow on 9 February 1934. He spent his childhood in the town of Pushkino near Moscow, where he graduated from school with a gold medal. Almost the entire life of Victor Nikolaevich was connected with the Lomonosov Moscow State University (MSU) and specifically with the Department of Higher Algebra of the Faculty of Mechanics and Mathematics (Mech-Mat). He enrolled at Mech-Mat in 1953, graduated from it in 1958 to become a PhD student at the Department of Algebra, and started working at this Department in 1961. The outstanding algebraist Anatolii Illarionovich Shirshov was his research advisor. Experts in the theory of associative and Lie rings are well aware of the results of Shirshov,