Infinite Dimensional Linear Groups Research Articles

On non-Abelian infinite-dimensional linear groups

We study non-Abelian solvable infinite-dimensional linear groups of infinite $p$-rank ($р \geqslant 0$) and of infinite fundamental dimensionality, whose any proper non-Abelian subgroup of infinite $p$-rank has finite fundamental dimensionality. We obtain the description of structure of groups from this class.

On some topics in the theory of infinite dimensional linear groups

In this paper we present a synopsis of some recent results concerned with infinite dimensional liner groups, including generalizations of irreducibility, the central dimension of a linear group, groups with finite dimensional orbits and the maximal and minimal conditions on subgroups of infinite central dimension.

On an analogue of a theorem of P. Hall for infinite dimensional linear groups

Let A be a vector space over a field F and let G be a subgroup of $$\mathrm{GL}(F,A)$$. This paper gives some further linear variations of a well-known theorem due to Philip Hall. For example we prove that if F has characteristic p and B is a finite dimensional subspace of A of dimension d such that A / B is G-hypercentral, and if G has finite section p-rank r, then the upper G-hypercenter has finite codimension in A, bounded by a function of d, r only. Analogues of these results in characteristic 0 are also obtained.

On the structure of some infinite dimensional linear groups

ABSTRACTIf G is a group and if the upper hypercenter, Z, of G is such that G∕Z is finite then a recent theorem shows that G contains a finite normal subgroup L such that G∕L is hypercentral. The purpose of the current paper is to obtain a version of this result for subgroups G of GL(F,A), when A is an infinite dimensionalF-vector space.

Infinite dimensional linear groups with many G — invariant subspaces

Abstract Let F be a field, A be a vector space over F, GL(F, A) be the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dimF(BFG/B) is finite. A subspace B is called almost G-invariant, if dim F(B/Core G(B)) is finite. In the current article, we study linear groups G such that every subspace of A is either nearly G-invariant or almost G-invariant in the case when G is a soluble p-group where p = char F.

On some infinite dimensional linear groups

Abstract Let F be a field, A be a vector space over F, and GL(F,A) the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dimF(BFG/B) is finite. A subspace B is called almost G-invariant, if dimF(B/CoreG(B)) is finite. In the present article we begin the study of subgroups G of GL(F,A) such that every subspace of A is either nearly G-invariant or almost G-invariant. More precisely, we consider the case when G is a periodic p′-group where p = charF.

Solvable infinite-dimensional linear groups with restrictions on the nonabelian subgroups of infinite rank

Under study are the solvable nonabelian linear groups of infinite central dimension and sectional p-rank, p ≥ 0, in which all proper nonabelian subgroups of infinite sectional p-rank have finite central dimension. We describe the structure of the groups of this class.

Locally soluble infinite-dimensional linear groups with restrictions on nonAbelian subgroups of infinite ranks

We are concerned with locally soluble linear groups of infinite central dimension and infinite sectional p-rank, p ⩾ 0, in which every proper non-Abelian subgroup of infinite sectional p-rank has finite central dimension. It is proved that such groups are soluble.

Infinite-dimensional linear groups with restrictions on subgroups that are not soluble A 3-groups

We are concerned with infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble A3-groups and all of whose proper subgroups, which are not soluble A3-groups, have finite central dimension. The structure of groups in this class is described. The case of infinite-dimensional locally nilpotent linear groups satisfying the specified conditions is treated separately. A similar problem is solved for infinite-dimensional locally soluble linear groups of infinite fundamental dimension that are not soluble A3-groups and all of whose proper subgroups, which are not soluble A3-groups, have finite fundamental dimension.

Linear groups with the minimal condition on subgroups of infinite central dimension

The authors study infinite dimensional linear groups with the minimal condition on subgroups of infinite central dimension.

Linear groups with the minimal condition on subgroups of infinite central dimension

The authors study infinite dimensional linear groups with the minimal condition on subgroups of infinite central dimension.

Generalisations of nilpotency of rings

In (5) and (6) we studied certain subgroups of infinite dimensional linear groups over rings. In particular we investigated how the structure of the subgroups was related to the structure of the rings over which the linear groups were defined. It became clear that it might prove useful to study generalised nilpotent properties of rings analogous to Baer nilgroups and Gruenberg groups. We look briefly at some classes of generalised nilpotent rings in this paper and obtain a lattice diagram exhibiting all the strict inclusions between the classes.