Subgroups of the full linear group over a semilocal ring

Let be a semilocal ring (a factor ring with respect to the Jacobson-Artin radical) for which the residue field C/m of its center C with respect to each maximal idealmC contains no fewer than seven elements. The structure of subgroups H in the full linear group GL(n, ) containing the group of diagonal matrices is considered. The main theorem: for any subgroup H there is a uniquely determined D-net of ideals such that G()HN(), whereN() is the normalizer of the D-net subgroup . A transparent classification of subgroups GL(n, ) normalizable by diagonal matrices is thus obtained. Further, the factor groupN()/G() is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 32–34, 1978.     

Subgroups of the full linear group over a Dedekind ring

We study the subgroups of the full linear group GL(n, R) over a Dedekind ring R that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup H there exists a unique D-net of ideals of R such that, where E() is the subgroup generated by all transvections of the net subgroup G(). and is the normalizer of G(). The subgroup E() is normal in. To study the factor group we introduce an intermediate subgroup F(), E() F() G(). The group is finite and is connected with permutations in the symmetric group. The factor group G()/F() is Abelian — these are the values of a certain determinant. In the calculation of F()/E() appears the SK₁-functor. Results are stated without proof.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 13–20, 1979.     

Modular group rings of the finitary symmetric group

The paper discusses approaches to constructing two-sided ideals of the modular group algebra of finitary symmetric group. In memory of Professor S. A. Amitsur     

Free products of finitary linear groups

In this note we show that the free product of any family of groups which are finitary linear over fields of the same characteristic , is still finitary linear over a field of characteristic .

     

Sylow subgroups of finitary linear groups

In this paper we describe the structure and the conjugacy classes of Sylow p-subgroups of FGL(V, ), the group of finitary -automorphisms of the -vector space V.The Author is member of the GNSAGA.     

Free products of finitary skew linear groups

We give two very different proofs of the following result. Let {G _γ:λɛΛ : λ ∈ Λ} be a family of finitary skew linear groups of the same characteristic p ≧ 0. Then the free product of the G _γ is isomorphic to some finitary skew linear group of characteristic p. This extends recent work of R. J. H. Minty on the skew linear case and of O. Puglisi on the finitary linear case.     

On finitary linear groups with a locally nilpotent maximal subgroup

We prove that a group G of finitary permutations, containing a locally nilpotent maximal subgroup M is locally solvable if M is not a 2-group. We also prove that the same is true if G is a periodic, non-modular, finitary linear group.     

Automorphisms of a finitary factor power of an infinite symmetric group

We consider a semigroup FP_\textfin⁺ ( \mathfrakS_\textfin( \mathbbN ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) defined as a finitary factor power of a finitary symmetric group of countable order. It is proved that all automorphisms of FP_\textfin⁺ ( \mathfrakS_\textfin( \mathbbN ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) are induced by permutations from \mathfrakS( \mathbbN ) \mathfrak{S}\left( \mathbb{N} \right) .     

Subgroups of the full linear group over the field of quotients of a principal ideal domain

The structure is described of all subgroups of the full linear group over the field of quotients of a principal ideal domain R which contain the special linear group over R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 86, pp. 185–187, 1979.     

Confined subgroups in periodic simple finitary linear groups

A subgroupX of the locally finite groupG is said to beconfined, if there exists a finite subgroupF≤G such thatX ^g∩F≠1 for allg∈G. Since there seems to be a certain correspondence between proper confined subgroups inG and non-trivial ideals in the complex group algebra ℂG, we determine the confined subgroups of periodic simple finitary linear groups in this paper. Dedicated to the memory of our friend and collaborator Richard E. Phillips