Primitive group rings of soluble groups

Some primitive group rings

‎The main part of this article is a survey on torsion subgroups of the unit group of‎ ‎an integral group ring‎. ‎It contains the major parts of my talk given at the‎ ‎conference‎ ‎Groups,‎ ‎Group Rings and Related Topics‎ ‎at UAEU in Al Ain October 2013‎. ‎In the second part special emphasis is layed on $p$‎ - ‎subgroups and on the‎ ‎open question whether there is a Sylow like theorem in the‎ ‎normalized unit group of an integral group ring‎. ‎For specific classes of finite groups we prove that $p$‎ - ‎subgroups‎ ‎of the normalized unit group of its integral group rings $V(mathbb{Z}G)$ ‎are isomorphic to subgroups of $G‎ .‎$ In particular for $p = 2$ this is shown ‎when $G$ has abelian Sylow $2$‎ - ‎subgroups‎. ‎This extends results known‎ ‎for soluble groups to classes of groups which are not contained in the‎ ‎class of soluble groups‎.

Let K [G] denote the group ring of G over the field K. Until recently it had been an open question as to whether K[G] could be primitive, that is have a faithful irreducible module, if G <1>. An affirmative answer has just been given in the important paper of E. Formanek and R. L. Snider where a large number of examples of primitive group rings were exhibited. In this paper we continue this study.

This thesis records an attempt to prove the two conjecture: Conjecture A: Every finite non-regular primitive permutation group of degree n contains permutations fixing one point but fixing at most $n^{1/2}$ points. Conjecture C: Every finite irreducible linear group of degree m &gt; 1 contains an element whose fixed-point space has dimension at most m/2. Variants of these conjectures are formulated, and C is reduced to a special case of A. The main results of the investigation are: Theorem 2: Every finite non-regular primitive permutation group of degree n contains permutations which fix one point but fix fewer than (n+3)/4 points. Theorem 3: Every finite non-regular primitive soluble permutation group of degree n contains permutations which fix one point but fix fewer than $n^{7/18}$ points. Theorem 4: If H is a finite group, F is a field whose characteristic is 0 or does not divide the order of H, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than m/2. Theorem 5: If H is a finite soluble group, F is any field, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than 7m/18. Proofs of these assertions are to be found in Chapter II; examples which show the limitations on possible strenghtenings of the conjectures and results are marshalled in Chapter III. A detailed formulation of the problems and results is contained in section 1.

Let R be a ring and let G be a soluble group. In this situation we shall give necessary and sufficient conditions for RG to have a right Artinian right quotient ring. In the course of this work, we shall also consider the Goldie rank problem for soluble groups and record an affirmative answer to the zero divisor conjecture for soluble groups.

Let G be a subgroup of the group GL ( V, F ) of automorphisms of a vector space V over a field F . By definition, the augmentation dimension of G , augdim F G for short, is the F -dimension of the vector subspace V (ω FG ), where ω FG is the augmentation ideal of the group ring FG . It is well known that G is finitary if and only if every finitely generated subgroup of G has finite augmentation dimension. In the antipodes of this concept, we define G to be antifinitary , if every proper subgroup of G of infinite augmentation dimension is finitely generated. In this paper we describe some generalized soluble antifinitary groups. 2000 Mathematics Subject Classification: 20F22; 20H20.