On subgroups of finite index in compact Hausdorff groups

Let G be a compact Hausdorff group and n a positive integer. It is proved that all subnormal subgroups of G of index dividing n are open if and only if there are only finitely many such subgroups, and that all subgroups of finite index in G are open if and only if there are only countably many such subgroups.     

The influence of S-quasinormality of some subgroups of prime power order on the structure of finite groups

Let G be a finite group. Two subgroups H and K of G are said to permute if áH,K? = HK = KH\langle H,K\rangle = HK = KH. A subgroup H of G is S-quasinormal in G if it permutes with every Sylow subgroup of G. In this paper we investigate the influence of S-quasinormality of some subgroups of prime power order of a finite group on its supersolvability.     

Finite groups in which normality is a transitive relation

Let G be a finite group. We say that G is a T₀-group, if its Frattini quotient group G/F(G)G/\Phi (G) is a T-group, where by a T-group we mean a group in which every subnormal subgroup is normal. We determine the structure of a non T₀-group G all of whose proper subgroups are T₀-groups.     

Product of nilpotent subgroups

We will say that a subgroup X of G satisfies property C in G if C_G(X?X^g)\leqq X?X^g{\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}} for all g ? G{g}\in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if N\triangleleft GN\triangleleft G is nilpotent and X is a nilpotent subgroup of G then C_G(N?X)\leqq XC_G(N\cap X)\leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.     

On finitary linear groups with a locally nilpotent maximal subgroup

We prove that a group G of finitary permutations, containing a locally nilpotent maximal subgroup M is locally solvable if M is not a 2-group. We also prove that the same is true if G is a periodic, non-modular, finitary linear group.     

Profinite groups with many commuting pairs or involutions

We prove that if the set of commuting pairs of a profinite group G has positive Haar measure then G is abelian by finite. Using this we show that the set I of involutions has positive measure exactly if I contains a nonempty open subset of G.     

On self-normality and abnormality in the alternating group Ap

An old problem proposed by Huppert, Doerk and Hawkes motivates us to investigate the relationship between an abnormal subgroup and self-normalizing in non-solvable groups. A subgroup H of a group G is called second maximal if H is maximal in all maximal subgroups of G containing H. Our result is that if H is a second maximal subgroup of the alternating group A_p of prime degree, then H is abnormal in A_p if and only if H is self-normalizing.     

Finite groups with trivial Frattini subgroup

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.     

On permutable subgroups of finite groups

Let \frak Z \frak Z be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, \frak Z \frak Z contains exactly one and only one Sylow p-subgroup of G. A subgroup H of a finite group G is said to be \frak Z \frak Z -permutable if H permutes with every member of \frak Z \frak Z . The purpose here is to study the influence of \frak Z \frak Z -permutability of some subgroups on the structure of finite groups. Some recent results are generalized.     

Permutability of subgroups of G×H that are direct products of subgroups of the direct factors

If M and S are two subgroups of a group G, M and S permute if MS = SM. Furthermore, M is a permutable subgroup of G if M permutes with every subgroup of G. We give necessary and sufficient conditions for M, a subgroup of G, to permute with a subgroup of G 2 H given that G and H are finite groups. The main part of the paper involves the development of a characterization of permutable subgroups of G 2 H that are direct products of subgroups of the direct factors; that is, subgroups that are equal to A 2 B where A \leqq \leqq G and B \leqq \leqq H.     

The Center of a Regular Algebraic Monoid

This paper is devoted to proving the following result: Let M be an irreducible regular algebraic monoid with unit group G. Then the center of M is unit regular and coincides with the closure of the center of G. Moreover if the M is reductive, then its center also coincides with the intersection of all maximal diagonizable submonoids of M.     

Group-invariant Percolation on Graphs

Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new masstransport technique that has been occasionally used elsewhere and is developed further here.¶ Perhaps surprisingly, these investigations of group-invariant percolation produce results that are new in the Bernoulli setting. Most notably, we prove that critical Bernoulli percolation on any nonamenable Cayley graph has no infinite clusters. More generally, the same is true for any nonamenable graph with a unimodular transitive automorphism group.¶ We show that G is amenable if for all $ \alpha < 1 $ \alpha < 1 , there is a G-invariant site percolation process w \omega on X with $ {\bf P} [x \in \omega] > \alpha $ {\bf P} [x \in \omega] > \alpha for all vertices x and with no infinite components. When G is not amenable, a threshold $ \alpha < 1 $ \alpha < 1 appears. An inequality for the threshold in terms of the isoperimetric constant is obtained, extending an inequality of Häggström for regular trees.¶ If G acts transitively on X, we show that G is unimodular if the expected degree is at least 2 in any G-invariant bond percolation on X with all components infinite.¶ The investigation of dependent percolation also yields some results on automorphism groups of graphs that do not involve percolation.     

On p-chief factors and extensions of $ \mathbb{K} $ G-modules

The subject of this paper is the relationship between the set of chief factors of a finite group G and extensions of an irreducible \mathbbK \mathbb{K} G-module U ( \mathbbK \mathbb{K} a field). Let H / L be a p-chief factor of G. We prove that, if H / L is complemented in a vertex of U, then there is a short exact sequence of Ext-functors for the module U and any \mathbbK \mathbb{K} G-module V. In some special cases, we prove the converse, which is false in general. We also consider the intersection of the centralizers of all the extensions of U by an irreducible module and provide new bounds for this group.     

D'Alembert's functional equations on metabelian groups

Summary. We extend d'Alembert's classical functional equation by replacing the domain of definition Bbb R {Bbb R} of the solutions by a metabelian group G and simultaneously replacing the group involution by an arbitrary involution of G. We find all complex valued solutions. In particular we show that the continuous solutions have the same form as in the abelian case if G is connected.     

Idempotents in a complex group algebra, projective modules, and the von Neumann algebra

The complex group algebra \Bbb CG{\Bbb C}G of a countable group G can be imbedded in the von Neumann algebra NG of G. If G is torsion-free, and if P is a finitely generated projective module over \Bbb CG{\Bbb C}G it is proved that the central-valued trace of NG?_{\Bbb CG}PNG\otimes _{{\Bbb C}G}P, i.e. of an idempotent \Bbb CG{\Bbb C}G-matrix A defining P is equal to the canonical trace k(P)\kappa (P) times identity I. It follows that k(P)\kappa (P) characterizes the isomorphism type of NG?_{\Bbb CG}PNG\otimes _{{\Bbb C}G}P.¶If k(P)\kappa (P) is an integer, e.g., if the weak Bass conjecture holds for G then NG?_{\Bbb C G}PNG\otimes _{{\Bbb C} G}P is free. It is also shown that for certain classes of groups geometric arguments can be used to prove the Bass conjecture.     

Group Connectivity of 3-Edge-Connected Chordal Graphs

Let A be a finite abelian group and G be a digraph. The boundary of a function f: E(G)ZA is a function ‘f: V(G)ZA given by ‘f(v)=~_{e leaving v}f(e)m~_{e entering v}f(e). The graph G is A-connected if for every b: V(G)ZA with ~_{v] V(G)} b(v)=0, there is a function f: E(G)ZA{0} such that ‘f=b. In [J. Combinatorial Theory, Ser. B 56 (1992) 165-182], Jaeger et al showed that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̈́. It is conjectured that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̓ and that every 5-edge-connected graph is A-connected, for every abelian group A with |A|́.¶ In this note, we investigate the group connectivity of 3-edge-connected chordal graphs and characterize 3-edge-connected chordal graphs that are A-connected for every finite abelian group A with |A|́.     

Non-Commutative Ribbons and Quasi-Differential Operators

This paper is devoted to a non-commutative generalization of a classical result occurring in the context of the modular representation theory of the symmetric group. We prove that a non-commutative Schur ribbon function R_I is annihilated by the quasi-differential operator D_Pk if and only if the composition I is the external border of a k-core.     

Glatte p-Gruppen

A p-group G is called smooth if there is a smooth chief series in G, that is a chief series in which any two factor groups of the same order are isomorphic. It is shown that in most cases non-abelian smooth p-groups are of nilpotency class two and that a smooth p-group is not much larger than the largest abelian group in its smooth chief series.     

On the “three-space” problem for locally quasi-convex topological groups

We study, in the context of abelian topological groups, the "three-space" problem for the property of being locally quasi-convex, after a paper of M. Bruguera. Our main contributions are: establishing a 3-lemma suitable to work with topological groups (which allows to translate the basic elements of homological algebra to the category of topological groups) and obtaining the analogue, for topological groups, of Dierolf's result in topological vector spaces:¶Theorem. Given two abelian locally quasi-convex groups H and G there exists a non-locally-quasi-convex extension of H and G if and only if there exists a non-locally-quasi-convex extension of S (the circle group) and G.     

Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space

We establish a close link between the amenability property of a unitary representation p \pi of a group G (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system (\Bbb S_p, G) ({\Bbb S}_{\pi}, G) , where \Bbb S_H {\Bbb S}_{\cal H} is the unit sphere the Hilbert space of representation. We prove that p \pi is amenable if and only if either p \pi contains a finite-dimensional subrepresentation or the maximal uniform compactification of (\Bbb S_p ({\Bbb S}_{\pi} has a G-fixed point. Equivalently, the latter means that the G-space (\Bbb S_p, G) ({\Bbb S}_{\pi}, G) has the concentration property: every finite cover of the sphere \Bbb S_p {\Bbb S}_{\pi} contains a set A such that for every e > 0 \epsilon > 0 the e \epsilon -neighbourhoods of the translations of A by finitely many elements of G always intersect. As a corollary, amenability of p \pi is equivalent to the existence of a G-invariant mean on the uniformly continuous bounded functions on \Bbb S_p {\Bbb S}_{\pi} . As another corollary, a locally compact group G is amenable if and only if for every strongly continuous unitary representation of G in an infinite-dimensional Hilbert space H {\cal H} the system (\Bbb S_H, G) ({\Bbb S}_{\cal H}, G) has the property of concentration.