Nontrivial Normal Subgroup Research Articles

The Atiyah Conjecture and Artinian Rings

Let G be a group such that its finite subgroups have bounded order, let d denote the lowest common multiple of the orders of the finite subgroups of G, and let K be a subfield of C that is closed under complex conjugation. Let U(G) denote the algebra of unbounded operators affiliated to the group von Neumann algebra N(G), and let D(KG,U(G)) denote the division closure of KG in U(G); thus D(KG,U(G)) is the smallest subring of U(G) containing KG that is closed under taking inverses. Suppose n is a positive integer, and \alpha \in \Mat_n(KG). Then \alpha induces a bounded linear map \alpha: l^2(G)^n \to \l^2(G)^n, and \ker\alpha has a well-defined von Neumann dimension \dim_{N(G)} (\ker\alpha). This is a nonnegative real number, and one version of the Atiyah conjecture states that d \dim_{N(G)}(\ker\alpha) \in Z. Assuming this conjecture, we shall prove that if G has no nontrivial finite normal subgroup, then D(KG,U(G)) is a d \times d matrix ring over a skew field. We shall also consider the case when G has a nontrivial finite normal subgroup, and other subrings of U(G) that contain KG.

On normal subgroups of Aut k (A 1(k)), the k-automorphisms group of Weyl algebra A 1(k)

Let A1(k) = k[x,y] with the relation yx−xy = 1, be the first Weyl algebra over a field k. It is well known that the group G : = Autk(A1(k)) of the k-automorphisms of A1(k) is an amalgamate group with two of its subgroups (see Alev in Commun Algebra 14(8):1365–1378, 1986), so G must have non-trivial normal subgroups. But none of such subgroup is known. When the field k is a finite set, we reveal in this paper one non trivial normal subgroup (may be the first) of G.

Linear representations of soluble groups of finite Morley rank

Sufficient conditions are given for groups of finite Morley rank having nontrivial torsion-free nilpotent normal subgroups to have linear representations with small kernels. In particular, centreless connected soluble groups of finite Morley rank with torsion-free Fitting subgroups have faithful linear representations. Along the way, using a notion of definable weight space, we prove that certain connected soluble groups of finite Morley rank with torsion-free derived subgroup can be embedded in groups of finite Morley rank whose Fitting subgroups have definable abelian supplements.

Subgroups of finite index and the just infinite property

A residually finite (profinite) group G is just infinite if every non-trivial (closed) normal subgroup of G is of finite index. This paper considers the problem of determining which (closed) subgroups of finite index of a just infinite group are themselves just infinite. If G is just infinite and not virtually abelian, we show that G is hereditarily just infinite if and only if all maximal (closed) subgroups of finite index are just infinite. This result will be used to show that a finitely generated pro- p group G is just infinite if and only if G has no non-trivial finite normal subgroups and Φ ( G ) has a just infinite open subgroup.

Normal automorphisms of relatively hyperbolic groups

An automorphism $\alpha$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we show that for any such group $G$, $Inn(G)$ has finite index in the subgroup $Aut_n(G)$ of normal automorphisms. If, in addition, $G$ is non-elementary and has no finite non-trivial normal subgroups, then $Aut_n(G)=Inn(G)$. As an application, we show that $Out(G)$ is residually finite for every finitely generated residually finite group $G$ with infinitely many ends.

On nilpotent subgroups containing non-trivial normal subgroups

Let G be a non-trivial finite group and let A be a nilpotent subgroup of G. We prove that if jG : Ajc expðAÞ, the exponent of A, then A contains a non-trivial normal subgroup of G. This extends an earlier result of Isaacs, who proved this in the case where A is abelian. We also show that if the above inequality is replaced by jG : Aj < ExpðGÞ, where ExpðGÞ denotes the order of a cyclic subgroup of G with maximal order, then A contains a non-trivial characteristic subgroup of G. We will use these results to derive some facts about transitive permutation groups.

On pro-p analogues of limit groups via extensions of centralizers

We begin a study of a pro-p analogue of limit groups via extensions of centralizers and call \({\mathcal{L}}\) this new class of pro-p groups. We show that the pro-p groups of \({\mathcal{L}}\) have finite cohomological dimension, type FP∞ and non-positive Euler characteristic. Among the group theoretic properties it is proved that they are free-by-(torsion free nilpotent) and if non-abelian do not have a finitely generated non-trivial normal subgroup of infinite index. Furthermore it is shown that every 2 generated pro-p group in the class \({\mathcal{L}}\) is either free pro-p or abelian.

Some primitive linear groups of prime degree

A classical problem in finite group theory dating back to Jordan, Klein, E. H. Moore, Dickson, Blichfeldt etc. is to determine all finite subgroups in SL ( n , C ) up to conjugation for some small values of n . This question is important in group theory as well as in the study of quotient singularities. Some results of Blichfeldt when n = 3 , 4 were generalized to the case of finite primitive subgroups of SL ( 5 , C ) and SL ( 7 , C ) by Brauer and Wales. The purpose of this article is to consider the following case. Let p be any odd prime number and G be a finite primitive subgroup of SL ( p , C ) containing a non-trivial monomial normal subgroup H so that H has a non-scalar diagonal matrix. We will classify all these groups G up to conjugation in SL ( p , C ) by exhibiting the generators of G and representing G as some group extensions. In particular, see the Appendix for a list of these subgroups when p = 5 or 7.

Normal subgroups of infinite symmetric groups, with an application to stratified set theory

It is generally known that infinite symmetric groups have few nontrivial normal subgroups (typically only the subgroups of bounded support) and none of small index. (We will explain later exactly what we mean by small). However the standard analysis relies heavily on the axiom of choice. By dint of a lot of combinatorics we have been able to dispense—largely—with the axiom of choice. Largely, but not entirely: our result is that if X is an infinite set with ∣X∣ = ∣X × X∣ then Symm(X) has no nontrivial normal subgroups of small index. Some condition like this is needed because of the work of Sam Tarzi who showed [4] that, for any finite group G, there is a model of ZF without AC in which there is a set X with Symm(X)/FSymm(X) isomorphic to G.The proof proceeds in two stages. We consider a particularly useful class of permutations, which we call the class of flexible permutations. A permutation of X is flexible if it fixes at least ∣X∣-many points. First we show that every normal subgroup of Symm(X) (of small index) must contain every flexible permutation. This will be theorem 4. Then we show (theorem 7) that the flexible permutations generate Symm(X).

On Hyper (Abelian of Finite Rank) Groups

We study the class of groups G, each of whose non-trivial images contains a non-trivial abelian normal subgroup of finite rank. This is very much wider than the class, studied earlier by Robinson and others, of hyperabelian groups H with finite abelian section rank. Our main results are that these groups G are hypercentral by residually finite and are divisible by hypo(abelian of finite exponent). They need not be divisible by residually finite, unlike the groups H above. In practice, we work with a somewhat wider but less easily described class of groups.

Abstract simplicity of complete Kac–Moody groups over finite fields

Let G be a Kac–Moody group over a finite field corresponding to a generalized Cartan matrix A , as constructed by Tits. It is known that G admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac–Moody group G ̂ which is defined to be the closure of G in the automorphism group of its building. Our main goal is to determine when complete Kac–Moody groups are abstractly simple, that is have no proper non-trivial normal subgroups. Abstract simplicity of G ̂ was previously known to hold when A is of affine type. We extend this result to many indefinite cases, including all hyperbolic generalized Cartan matrices A of rank at least four. Our proof uses Tits’ simplicity theorem for groups with a BN-pair and methods from the theory of pro- p groups.

On 9- and 10-decomposable finite groups

A finite group G is called n-decomposable if every proper non-trivial normal subgroup of G is a union of n distinct conjugacy classes of G. In some research papers, the question of finding all positive integer n such that there is an n-decomposable finite group was posed.

Polynomial identities and maximal subgroups of skew linear groups

Suppose that D is a division ring with center F and N is a non-central normal subgroup of GLn(D). In this paper we generalize some known results about maximal subgroups of GLn(D) to maximal subgroups of N. More precisely, we prove that if M is a maximal subgroup of N such that F[M] satisfies a polynomial identity and \(C_{M_{n}(D)}(M) \backslash F\) contains an algebraic element over F or \(C_{M_{n}(D)}(M)=F\) and either n ≥ 2 or n = 1 and M is not abelian, then [D : F] < ∞.

Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups

We prove that if a countable discrete group Γ is w-rigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. $\Gamma=SL(2,\mathbb{Z})\ltimes\mathbb{Z}^2$ , or Γ=H×H’ with H an infinite Kazhdan group and H’ arbitrary), and $\mathcal{V}$ is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. $\mathcal{V}$ countable discrete, or separable compact), then any $\mathcal{V}$ -valued measurable cocycle for a measure preserving action $\Gamma\curvearrowright X$ of Γ on a probability space (X,μ) which is weak mixing on H and s-malleable (e.g. the Bernoulli action $\Gamma\curvearrowright[0,1]^{\Gamma}$ ) is cohomologous to a group morphism of Γ into $\mathcal{V}$ . We use the case $\mathcal{V}$ discrete of this result to prove that if in addition Γ has no non-trivial finite normal subgroups then any orbit equivalence between $\Gamma\curvearrowright X$ and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ≃Λ.

On orders and vanishing of integral cohomology groups

Let G be a finite group with a non-trivial normal subgroup N such that G / N is cyclic. In this paper we obtain a recurrence relation involving the orders of the integral cohomology groups H n ( G , Z ) , n ⩾ 1 . We then use this result to show that no two consecutive integral cohomology groups of G can vanish. This in turn is used to show that if G is a finite group and k is a field of characteristic p such that H 1 ( G , k ) ≠ 0 , then H 2 n ( G , k ) ≠ 0 for all n ⩾ 0 . Some results relating the restriction and corestriction maps in integral cohomology are also obtained.

On the product of a nilpotent group and a group with non-trivial center

It is proved that a finite group G = A B which is a product of a nilpotent subgroup A and a subgroup B with non-trivial center contains a non-trivial abelian normal subgroup.

Three amalgams with remarkable normal subgroup structures

We construct three groups Λ 1 , Λ 2 , Λ 3 , which can all be decomposed as amalgamated products F 9 ∗ F 81 F 9 and have very few normal subgroups of finite or infinite index. Concretely, Λ 1 is a simple group, Λ 2 is not simple but has no non-trivial normal subgroup of infinite index, and Λ 3 is not simple but has no proper subgroup of finite index.

Free product decompositions in images of certain free products of groups

Let F be the free product of n groups and let R be a normal subgroup generated (as a normal subgroup) by m elements of F, where m < n . The Main Theorem gives sufficient conditions for families of fewer than n − m subgroups in certain quotients of F / R to generate their free product. This leads to a more direct proof of a result of the first author, that if G is a group having a presentation with n generators and m relators, where m < n , then any generating set for G contains n − m elements that freely generate a free subgroup of G. Another consequence is that an n-generator one-relator group cannot be generated by fewer than n − 1 subgroups each having a non-trivial abelian normal subgroup.

Relatively hyperbolic groups are [formula omitted]-simple

We characterize relatively hyperbolic groups whose reduced C ∗ -algebra is simple as those, which have no non-trivial finite normal subgroups.

A characterization of the full wreath product

A groupoid [3, 17] is a set Q endowed with a binary product, that is, a map from Q × Q to Q. In his 1964 paper [7], Bernd Fischer studied distributive quasigroups, which by definition are groupoids Q for which right multiplication by any fixed element gives an automorphism of Q as does left multiplication. Fischer proved that the right multiplication group R(Q) of a finite distributive quasigroup Q is solvable. He did this by showing that, for a minimal counterexample, the right multiplications T = {μa : g 7→ ga | a ∈ Q } are a generating conjugacy class of involutions in R(Q) ≤ Aut(Q) with the additional property that |tr| = 3 for distinct t and r from T . He then proved that this property forces finite R(Q) to have a normal 3-group of index 2. This led Fischer to consider [8, 9, 10] the extent to which finite symmetric groups can be characterized through being generated by a conjugacy class of involutions with all products of order 1, 2, or 3—a class of 3-transpositions, since the model is the transposition (2-cycle) class of Sym(Ω), the symmetric group on the set Ω. In a landmark theorem [10], Fischer found all finite 3transposition groups with no nontrivial solvable normal subgroups, discovering three new sporadic simple groups along the way. At the same time that Fischer was considering distributive quasigroups, George Glauberman was working on certain special groupoids, called Bruck loops. Glauberman [13] proved that finite Bruck and finite Moufang loops of odd order are solvable. His approach was similar to Fischer’s. He constructed a canonical conjugacy class T of involutory loop permutations with the additional property that |tr| was always odd for t and r from T . In his famous Z∗-theorem [14], Glauberman then proved that a finite group generated by such a class T has a normal subgroup of odd order and index 2 (a result also proved by Fischer [7] in the special case where all orders |tr| are powers of some fixed odd prime). Fischer’s and Glauberman’s work on finite quasigroups and loops had a profound effect on the theory of finite simple groups. For a normal set of involutions