Abelian Subgroup Of Finite Index Research Articles

Infinite groups whose group algebras satisfy the converse of Schur’s lemma

In this work, we give some necessary and/or sufficient conditions for a group algebra of infinite group to satisfy the converse of Schur’s Lemma. Many classes of groups are investigated such as abelian groups, hypercentral groups, groups having abelian subgroup of finite index and finitely generated soluble groups.

Groups with bounded centralizer chains and the Borovik–Khukhro conjecture

Abstract Let G be a locally finite group and let F ⁢ ( G ) {F(G)} be the Hirsch–Plotkin radical of G. Let S denote the full inverse image of the generalized Fitting subgroup of G / F ⁢ ( G ) {G/F(G)} in G. Assume that there is a number k such that the length of every nested chain of centralizers in G does not exceed k. The Borovik–Khukhro conjecture states, in particular, that under this assumption, the quotient G / S {G/S} contains an abelian subgroup of finite index bounded in terms of k. We disprove this statement and prove a weak analogue of it.

Large characteristic subgroups with modular subgroup lattice

It is known that if a group contains an abelian subgroup of finite index, then it also has an abelian characteristic subgroup of finite index. The aim of this paper is to prove that corresponding results hold when abelian subgroups are replaced either by subgroups having a modular subgroup lattice or by quasihamiltonian subgroups.

A Note on Large Characteristic Subgroups

ABSTRACTIt is known that if a group G has an abelian subgroup of finite index n, then it contains an abelian characteristic subgroup A of index at most nn. The aim of this paper is to improve this bound by showing that the characteristic subgroup A can be chosen of index at most n2. Examples prove that this bound is the best possible. Our main result is obtained as an application of a general method for the construction of large characteristic subgroups.

Some remarks on symmetrical extensions of graphs

It is proved that results from a previous paper of the author on symmetrical 2-extensions of graphs can be extended to symmetrical p-extensions of graphs for any prime p. In particular, it is proved that, for any prime p, there are only finitely many symmetrical p-extensions of a locally finite graph with an abelian subgroup of finite index in its automorphism group. Some refinements of these results are also obtained. In addition, we consider the question on the possibility to represent symmetrical extensions of a d-dimensional grid (and similar graphs) in the d-dimensional affine Euclidean space in such a way that a corresponding vertex-transitive group of automorphisms of the extension is induced by some crystallographic group of motions of the space.

The finiteness of the number of symmetrical 2-extensions of the d-dimensional grid and similar graphs

We prove that there are only finitely many symmetrical 2-extensions of a locally finite graph whenever the automorphism group of the graph has an abelian subgroup of finite index (this case is of interest for certain applications). Some refinements and generalizations of this result are also obtained.

A note on groups of infinite rank whose proper subgroups are abelian-by-finite

It is proved that if $G$ is a locally (soluble-by-finite) group of infinite rank in which every proper subgroup of infinite rank contains an abelian subgroup of finite index, then all proper subgroups of $G$ are abelian-by-finite.

Writing units of integral group rings of finite abelian groups as a product of Bass units

We give a constructive proof of the theorem of Bass and Milnor saying that if $G$ is a finite abelian group then the Bass units of the integral group ring $\Z G$ generate a subgroup of finite index in its units group $\U(\Z G)$. Our proof provides algorithms to represent some units that contribute to only one simple component of $\Q G$ and generate a subgroup of finite index in $\U(\Z G)$ as product of Bass units. We also obtain a basis $B$ formed by Bass units of a free abelian subgroup of finite index in $\U(\Z G)$ and give, for an arbitrary Bass unit $b$, an algorithm to express $b^{\varphi(|G|)}$ as a product of a trivial unit and powers of at most two units in this basis $B$.

On the spectral theory of groups of affine transformations of compact nilmanifolds

Let N be a connected and simply connected nilpotent Lie group, � a lattice in N, and �\N the corresponding nilmanifold. Let Aff(�\N) be the group of affine transformations of �\N. We characterize the countable subgroups H of Aff(�\N) for which the action of H on �\N has a spectral gap, that is, such that the associated unitary representation U 0 of H on the space of functions from L 2 (�\N) with zero mean does not weakly contain the trivial representation. Denote by T the maximal torus factor associated to �\N. We show that the action of H on �\N has a spectral gap if and only if there exists no proper H-invariant subtorus S of T such that the projection of H on Aut(T/S) has an abelian subgroup of finite index. We first establish the result in the case where �\N is a torus. In the case of a general nilmanifold, we study the asymptotic behaviour of matrix coefficients ofU 0 using decay properties of metaplectic representations of symplectic groups. The result shows that the existence of a spectral gap for subgroups of Aff(�\N) is equivalent to strong ergodicity in the sense of K. Schmidt. Moreover, we show that the action of H on �\N is ergodic (or strongly mixing) if and only if the corresponding action of H on T is ergodic (or strongly mixing).

WEAK AMENABILITY OF CONVOLUTION BANACH ALGEBRAS ON COMPACT HYPERGROUPS

In this paper we find necessary and sufficient conditions for weak amenability of the convolution Banach algebras A(K) and L2(K) for a compact hypergroup K, together with their applications to convolution Banach algebras Lp(K) (2 ≤ p < ∞). It will further be shown that the convolution Banach algebra A(G) for a compact group G is weakly amenable if and only if G has a closed abelian subgroup of finite index.

Abelian automorphisms groups with a finite number of orbits

Let G be a group and A a group of automorphisms of G . An A -orbit of G is a set of the form { g α | α ∈ A } , where g is an element of G . The aim of this paper is to prove that if A is abelian and G is a union of a finite number of A -orbits then G admits a normal abelian subgroup of finite index. This result answers affirmatively a question raised by Neumann and Rowley (1998) in [4] .

Convolutions on compact groups and Fourier algebras of coset spaces

We study two related questions. (1) For a compact group $G$, what are the ranges of the convolution maps on $\mathrm{A}(G\times G)$ given for $u,v$ in $\mathrm{A}(G)$ by $u\times v\mapsto u\ast\check{v}$ ($\check{v}(s)=v(s^{-1})$) and $u\times v\mapsto u

Biflatness and biprojectivity of the Fourier algebra

We show that the biflatness—in the sense of A. Ya. Helemskiĭ—of the Fourier algebra A(G) of a locally compact group G forces G to either have an abelian subgroup of finite index or to be non-amenable without containing \({\mathbb{F}}_{2}\) as a closed subgroup. An analogous dichotomy is obtained for biprojectivity.

The structure of groups whose subgroups are permutable-by-finite

AbstractA subgroup H of a group G is said to be permutable if HX = XH for each subgroup X of G, and the group G is called quasihamiltonian if all its subgroups are permutable. We shall say that G is a Q F-group if every subgroup H of G contains a subgroup K of finite index which is permutable in G. It is proved that every locally finite Q F-group contains a quasihamiltonian subgroup of finite index. In the proof of this result we use a theorem by Buckley, Lennox, Neumann, Smith and Wiegold concerning the corresponding problem when permutable subgroups are replaced by normal subgroups: if G is a locally finite group such that H/HG is finite for every subgroup H, then G contains an abelian subgroup of finite index.

Periodic groups whose simple modules have finite central endomorphism dimension

Theorem. If k k is an uncountable field and G G is a periodic group with no elements of order the characteristic of k k and if all simple k [ G ] k[G] modules have finite central endomorphism dimension, then G G has an abelian subgroup of finite index.

Amenability and weak amenability of the Fourier algebra

Let G be a locally compact group. We show that its Fourier algebra A(G) is amenable if and only if G has an abelian subgroup of finite index, and that its Fourier–Stieltjes algebra B(G) is amenable if and only if G has a compact, abelian subgroup of finite index. We then show that A(G) is weakly amenable if the component of the identity of G is abelian, and we prove some partial results towards the converse.

Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule

Let $\mathcal A$ be a dual Banach algebra with predual $\mathcal A_*$ and consider the following assertions: (A) $\mathcal A$ is Connes-amenable; (B) $\mathcal A$ has a normal, virtual diagonal; (C) $\mathcal A_*$ is an injective $\mathcal A$-bimodule. For general $\mathcal A$, all that is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B), and (C) are equivalent. We show that (C) always implies (B) whereas the converse is false for $\mathcal A = M(G)$ where $G$ is an infinite, locally compact group. Furthermore, we present partial solutions towards a characterization of (A) and (B) for $\mathcal A = B(G)$ in terms of $G$: For amenable, discrete $G$ as well as for certain compact $G$, they are equivalent to $G$ having an abelian subgroup of finite index. The question of whether or not (A) and (B) are always equivalent remains open. However, we introduce a modified definition of a normal, virtual diagonal and, using this modified definition, characterize the Connes-amenable, dual Banach algebras through the existence of an appropriate notion of virtual diagonal.

On the semisimplicity of polyhedral isometries

If a polyhedral complex K K has only finitely many isometry types of cells, then all of its cellular isometries are semisimple. If K K is 1-connected and non-positively curved, then any solvable group that acts freely by cellular isometries on K K is finitely generated and contains an abelian subgroup of finite index.

Geometric properties of some subspaces of 𝑉𝑁(𝐺)

Let G be a locally compact group. We show that any one of the spaces U C B ( G ^ ) , W A P ( G ^ ) , A P ( G ^ ) UCB(\hat G),WAP(\hat G),AP(\hat G) , and C δ ∗ ( G ) C_\delta ^ \ast (G) is Asplund if and only if the group G is finite. We also show that any one of the spaces V N ( G ) , U C B ( G ^ ) VN(G),UCB(\hat G) , and C δ ∗ ( G ) C_\delta ^\ast (G) has the DPP if and only if the group G has an abelian subgroup of finite index.

On a combinatorial problem in group theory

We say that a groupG ∈DS if for some integerm, all subsetsX ofG of sizem satisfy |X 2|<|X|2, whereX 2={xy|x,y ∈X}. It is shown, using a previous result of Peter Neumann, thatG ∈DS if and only if either the subgroup ofG generated by the squares of elements ofG is finite, orG contains a normal abelian subgroup of finite index, on which each element ofG acts by conjugation either as the identity automorphism or as the inverting automorphism.