We are concerned in this paper with the ideal structure of group rings of infinite simple locally finite groups over fields of characteristic zero, and its relation with certain subgroups of the groups, called confined subgroups. The systematic study of the ideals in these group rings was initiated by the second author in [15], although some results had been obtained previously (see [3, 1]). Let G be an infinite simple locally finite group and K a field of characteristic zero. It is expected that in most cases, the group ring KG will have the smallest possible number of ideals, namely three, (KG itself, {0} and the augmentation ideal), and this has been verified in some cases. In some interesting cases, however, the situation is different, and there are more ideals. We mention in particular the infinite alternating groups [3] and the stable special linear groups [9], in which the ideal lattice has been completely determined. The second author has conjectured that the presence of ideals in KG, other than the three unavoidable ones, is synonymous with the presence in the group of proper confined subgroups. Here a subgroup H of a locally finite group G is called confined, if there exists a finite subgroup F of G such that Hg∩F≠1 for all g∈G. This amounts to saying that F has no regular orbit in the permutation representation of G on the cosets of H.

Let R be a ring (always understood to be associative with a unit element 1). It is well known that an R-module is Noetherian if and only if all its submodules are finitely generated and that it has a finite composition series if and only if it is Noetherian and Artinian. This raises the question whether every finitely generated Artinian module is Noetherian; here it is enough to consider cyclic Artinian modules, by an induction on the length. This question has been answered (negatively) by Brian Hartley [5], who gives a construction of an Artinian uniserial module of uncountable composition-length over the group algebra of a free group of countable rank. If we are just interested in finding cyclic modules that are Artinian but not Noetherian, there is a very simple construction based on the fact that over a free algebra every countably generated Artinian module can be embedded in a cyclic module which is again Artinian. This is described in §2 below.

Let G be a connected semisimple group over an algebraically closed field K of characteristic p>0, and [gfr ]=Lie (G). Fix a linear function χ∈[gfr ]* and let [zfr ][gfr ](χ) denote the stabilizer of χ in [gfr ]. Set [Nscr ]p([gfr ]) ={x∈[gfr ][mid ]x[p]=0}. Let [Cscr ]χ([gfr ]) denote the category of finite-dimensional [gfr ]-modules with p-character χ. In [7], Friedlander and Parshall attached to each M∈Ob([Cscr ]χ([gfr ])) a Zariski closed, conical subset [Vscr ][gfr ](M)⊂[Nscr ]p([gfr ]) called the support variety of M. Suppose that G is simply connected and p is not special for G, that is, p≠2 if G has a component of type Bn, Cn or F4, and p≠3 if G has a component of type G2. It is proved in this paper that, for any nonzero M∈Ob([Cscr ]χ([gfr ])), the support variety [Vscr ][gfr ](M) is contained in [Nscr ]p([gfr ])∩[zfr ][gfr ](χ). This allows one to simplify the proof of the Kac–Weisfeiler conjecture given in [18].

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), we shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [8].

A negative answer to the Kuroš–Černikov Question 21 in [7], whether a group satisfying the normalizer condition is hypercentral, was given by Heineken and Mohamed in 1968 [6]. They constructed groups G satisfying:

(i) G is a locally finite p-group for a prime p,

(ii) G/G′≅Cp∞ and G′ is countable elementary abelian,

(iii) every proper subgroup of G is subnormal and nilpotent,

(iv) Z(G)={1},

(v) the set of normal subgroups of G contained in G′ is linearly ordered by set inclusion, see [3, p. 334],

(vi) KG′ is a proper subgroup in G for every proper subgroup K of G, see [6, Lemma 1(a)].

Let Fq be a finite field of characteristic r and order q=ra, V=Fnq the vector space of finite dimension n[ges ]1 over Fq, and Mat (n, q) the matrix algebra on V. We denote by Hn(q) (or Hn if q is clear from the context) one of the following classical groups: Sp (n, q) (with n even), the symplectic group on V; O(n, q) (with q odd), one of the full orthogonal groups on V in odd characteristic; U(n, q) (with q a square), the full unitary group on V.

Let the finite group A be acting on a finite group G with ([mid ]A[mid ], [mid ]G[mid ])=1. Let Γ be the semidirect product of A and G. Let χ be a character of Γ irreducible after restriction to G. In a previous paper by Brian Hartley and the author, we proved that the restriction of χ to S belongs to the set [Cscr ](S) obtained by running over all χ that arise in this manner, by assuming, in addition, that G is a product of extraspecial groups. This was proved in general, assuming only some condition on the Green functions of groups of Lie type that is not as yet fully verified. In the present paper, we define the map Q(χ): S[map ][Copf ] by Q(χ)(s) =[mid ]CG(s)[mid ]/χ(s). We prove that Q(χ)∈[Cscr ](S) under the same hypotheses. In particular, the character quotient Q(χ) is an ordinary character.

We show that given an affine algebraic group G over a field K and a finite subgroup scheme H of G there exists a finite dimensional G-module V such that V[mid ]H is free. The problem is raised in the recent paper by Kuzucuo˘glu and Zalesskiiˇ [15] which contains a treatment of the special case in which K is the algebraic closure of a finite field and H is reduced. Our treatment is divided into two parts, according to whether K has zero or positive characteristic. The essence of the characteristic 0 case is a proof that, for given n, there exists a polynomial GLn(ℚ)-module V of dimension 2nΠpp(n2), where the product is over all primes less than or equal to n+1, such that V is free as a ℚH-module for every finite subgroup H of GLn(ℚ). The module V is the tensor product of the exterior algebra Λ*(E), on the natural GLn(ℚ)-module E, and Steinberg modules Stp, one for each prime not exceeding n+1. The Steinberg modules also play the major role in the case in which K has characteristic p>0 and the key point in our treatment is to show that for a finite subgroup scheme H of a general linear group scheme (or universal Chevalley group scheme) G over K, the Steinberg module Stpn for G is injective (and projective) on restriction to H for n[Gt ]0. A curious consequence of this is that, despite the wild behaviour of the modular representation theory of finite groups, one has the following. Let H be a finite group and V a finite dimensional vector space. Then there exists a (well-understood) faithful rational representation π: GL(V)→GL(W) such that, for each faithful representation ρ: H→GL(V), the composite πορ: H→GL(W) is free, in particular all representations πορ are equivalent.

A stage in the classification of the periodic simple finitary linear groups by J. I. Hall and others, is a partial reduction to the countable case; specifically a simple group is isomorphic to a finitary linear group if and only if its countable subgroups are isomorphic to finitary linear groups. It is natural to ask, how common is it for a group to have a faithful finitary linear (or possibly only skew linear) representation, whenever each of its countable subgroups has such a representation? In very recent work F. Leinen and O. Puglisi have shown that periodic primitive finitary linear groups are countably recognizable, see [4]. It is easy to see that every abelian group is isomorphic to a finitary linear group, for example over the subfield of the complexes [Copf ] generated by all roots of unity. Thus the first non-obvious case, at the opposite extreme to the simple groups (and also in some sense to the primitive groups), is that of the nilpotent groups. We show the following, which, it should be noted, covers the case of periodic nilpotent groups.

In the nineteenth century, field theory brilliantly resolved a number of questions that had taxed mathematicians for centuries; for example, ‘The circle cannot be squared’ by straight edge and compass, and solving polynomial equations by radicals is not always possible. These successes have continued to be held up as superb examples of the power of mathematical thought, and are demonstrated at an undergraduate level. The purpose of this article is to provide another such natural example which leads to a concrete realisation of the free group on 2 generators.

We characterise the profinite groups with polynomial subgroup growth. We deduce that the property PSG is extension-closed in the category of all groups, and subgroup-closed in the category of finitely generated profinite groups.

If F is a subset of G⊆GL (n, [Kscr ])=GL(V, [Kscr ]) (where [Kscr ] is a field) the degree of F(=deg (F)) is the dimension of the [Kscr ]-space [V, F] spanned by

{v(f−1)[mid ]v∈V, f∈〈F〉};

note that in the special case F={g} we have [V, g]={v(g−1)[mid ]v∈V}. Our intention is to describe those irreducible linear groups G⊆GL (n, [Kscr ]) generated by elements whose degrees are small relative to n. To do this successfully it seems necessary to work within the restricted class of ‘solvable-by-locally finite’ groups (throughout, this class will be denoted by [Sscr ](L[Fscr ])). Somewhat surprisingly, it turns out that if G is an irreducible [Sscr ](L[Fscr ])-subgroup of GL (n, [Kscr ]) generated by elements of small degree (relative to n), then G has large non-abelian simple sections. For a linear group G, the restriction to the class [Sscr ](L[Fscr ]) is equivalent to insisting that G have no non-cyclic free subgroups (see [7, Section 5.6]). Our main result in this direction is the following structure theorem.

It is well known that the product of normal locally nilpotent (locally polycyclic) groups is again locally nilpotent (locally polycyclic). This is not true for locally solvable groups. We look at a class between locally polycyclic and locally solvable groups that is closed under normal products.

Fitting classes and their associated injectors have been considered in a number of classes of locally finite groups [9, 8, 4, 5]. In particular, Dixon has considered the class [Lfr ] of countable locally finite groups with min −p for all primes p. He has shown that if the [Lfr ]-group G is radical (that is, it has an ascending locally nilpotent series) then G has locally nilpotent injectors, any two of which are isomorphic. Question 7.4.7 in [5] asks whether the restriction of being radical can be removed in this result.

B. Hartley in [4] introduced the class of barely transitive groups. By definition, a group of permutations G on an infinite set X is called barely transitive if G itself is transitive on X while every orbit of every proper subgroup of G is finite. If G is locally finite and G′≠G then the theorem of B. Love, proved in [4], shows that G is a locally nilpotent p-group of Heineken–Mohamed type. However, it is not known if perfect barely transitive locally nilpotent p-groups exist. Obviously this is a more general question than the corresponding one about perfect minimal non FC-groups (see below for definition). In this work it will be shown that a barely transitive locally nilpotent p-group cannot be perfect if the stabilizer of a point is hypercentral and solvable.

1. Some groups with the small index property

In this paper we shall use the methods of the paper [6] by Hodges, Hodkinson, Lascar and Shelah to show that the free group of countably infinite rank and certain relatively free groups of countably infinite rank have the small index property.

Let G be a simple algebraic group over an algebraically closed field K of characteristic p with p[ges ]0. Then G contains finitely many conjugacy classes of unipotent elements. There is a unique class of unipotent elements of largest dimension – the class of regular unipotent elements; the centralizer of any such element is a unipotent group of dimension equal to the rank of G. For example, if G is the linear group SL (V) on the finite dimensional vector space V over K, then the regular unipotent elements are precisely the unipotent matrices consisting of a single Jordan block.

Throughout this paper G(k) denotes a Chevalley group of rank n defined over the field k, where n[ges ]3. Let Φ be the root system associated with G(k) and let Π={α1, α2, [eDDot ], αn} be a set of fundamental roots of Φ, with Φ+ being the set of positive roots of Φ with respect to Π. For α∈Π and γ∈Φ+, let nα(γ) be the coefficient of α in the expression of γ as a sum of fundamental roots; so γ=[sum ]α∈Πnα(γ)α. Also we recall that ht(γ), the height of γ, is given by ht(γ)=[sum ]α∈Πnα(γ). The highest root in Φ+ will be denoted by ρ. We additionally assume that the Dynkin diagram of G(k) is connected.

A 4n-dimensional Riemannian manifold (M, g) is hyperkähler if it possesses three anti-commuting complex structures I, J, K such that the metric g is Kähler with respect to each of them. The reduced holonomy group of such a manifold is necessarily a subgroup of Sp(n) so the Ricci tensor of g vanishes and (M, g) can be regarded as a positive definite solution to Einstein's equations in vacuum.