Isomorphisms of general linear groups over associative rings graded by an Abelian group

In this paper, we give a simpler proof of the Golubchik–Mikhalev–Zelmanov theorem on the structure of isomorphisms between general linear groups over associative rings, and also prove an extension of this theorem for linear groups over rings graded by an Abelian group.     

Isomorphisms of Steinberg groups over commutative rings

LetA andR be commutative rings, andm andn be integers3. It is proved that, if :St _m (A)St _n (R) is an isomorphism, thenm=n. Whenn4, we have: (1) Every isomorphism :St _n(A)St _n(R) induces an isomorphism:E _n (A)E _n (R), and is uniquely determined by; (2) IfSt _n (A) St _n (R) thenK _2.n (A)K _2.n (R); (3) Every isomorphismE _n (A) E _n (R) can be lifted to an isomorphismSt _n(A)St _n(R); (4)St _n(A) St _n(R) if and only ifAR. For the casen=3, ifSt ₃(A) andSt ₃(R) are respectively central extensions ofE ₃(A) andE ₃ (R), then the above (1) and (2) hold.The Project supported by National Natural Science Foundation of China     

Full subgroups of unitary groups over division rings

The subgroups, full of transvections, of unitary groups over division rings with an involution are described. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 193–200, 1987.     

Elementary equivalence of unitary linear groups over rings

We find necessary and sufficient conditions for two unitary linear groups over rings with involution with forms of hyperbolic rank at least 2 to be elementarily equivalent.     

Subnormal structure of non-stable unitary groups over rings

Let R be a commutative ring with identity in which 2 is invertible. Let H denote a subgroup of the unitary group U(2n,R,Λ) with n≥4. H is normalized by EU(2n,J,Γ^J) for some form ideal (J,Γ^J) of the form ring (R,Λ). The purpose of the paper is to prove that H satisfies a “sandwich” property, i.e. there exists a form ideal (I,Γ^I) such that

EU(2n,IJ⁸Γ^J,Γ)⊆H⊆CU(2n,I,Γ^I).     

Isomorphisms of the unitary group over a ring

We describe isomorphisms of unitary groups over associative rings with 1/2 that contain a compact system of idempotents. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 55–70, 2006.     

Structure of unitary groups over finite group rings and its application

In this paper, we determine all the normal forms of Hermitian matrices over finite group rings R = F_q²GR = {F_{{q^2}}}G, where q = p ^α , G is a commutative p-group with order p ^β . Furthermore, using the normal forms of Hermitian matrices, we study the structure of unitary group over R through investigating its BN-pair and order. As an application, we construct a Cartesian authentication code and compute its size parameters.     

The radical RN and the weakly solvable radical of linear groups over associative rings

This paper is devoted to the computation of the radicals RN and RN* and the weakly solvable radical for a number of basic classical linear groups over rings, including the unitary group over a ring with involution and matrix groups normalized by elementary Chevalley groups. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 31–115, 2007.     

Transvection parameter sets in linear groups over associative division rings and special Jordan algebras defined by these rings

In this paper we construct a correspondence between a class of irreducible linear groups of finite degree over an associative division ring D and special Jordan rings which are defined by D. Received: 14 September 2005     

Isomorphisms ofp-adic group rings

SupposeP is the ring ofp-adic integers,G is a finite group of orderp ⁿ , andPG is the group ring ofG overP. IfV _p (G) denotes the elements ofPG with coefficient sum one which are of order a power ofp, it is shown that the elements of any subgroupH ofV _p (G) are linearly independent overP, and if in additionH is of orderp ⁿ , thenPGPH. As a consequence, the lattice of normal subgroups ofG and the abelianization of the normal subgroups ofG are determined byPG.     

Isomorphisms of formal matrix incidence rings

Abstract—In a paper published in 2008 P. A. Krylov showed that formal matrix rings Ks(R) and Kt(R) are isomorphic if and only if the elements s and t differ up to an automorphism by an invertible element. Similar dependence takes place in many cases. In this paper we consider formal matrix rings (and algebras) which have the same structure as incidence rings. We show that the isomorphism problem for formal matrix incidence rings can be reduced to the isomorphism problem for generalized incidence algebras. For these algebras, the direct assertion of Krylov’s theorem holds, but the converse is not true. In particular, we obtain a complete classification of isomorphisms of generalized incidence algebras of order 4 over a field. We also consider the isomorphism problem for special classes of formal matrix rings, namely, formal matrix rings with zero trace ideals.