A lattice theoretic characterization of prefrattini subgroups

Consider an interval [H,G] in the lattice of subgroups of a finite soluble groupG. We define a certain set of subgroups in the lattice [H,G], and prove that they are conjugate inG. ForH=1 one gets the prefrattini subgroups ofG.     

The groups of the semigroup of compact subsets of a topological group

The compact subsets of a topological groupG form a semigroup,S(G), when multiplication is defined by set product. This semigroup is a topological semigroup when given the Vietoris topology. It would be expected that the subgroups ofS(G) should in some way be related to the groupG. This is the case. It is shown that the subgroups ofS(G) are both algebraically and topologically exactly the groups obtained as quotients of certain subgroups ofG. One consequence of this is that every subgroup ofS(G) is a topological group. Conditions are also given for these subgroups to be open or closed. Green's relations inS(G) have a particularly nice formulation. As a result, the relationsD andJ are equal inS(G). Moreover, the Schützenberger group of aD-class is a topological group that is topologically isomorphic to a quotient of certain subgroups ofG.     

C-Normality of Groups and Its Properties

A subgroupHis calledc-normal in groupGif there exists a normal subgroupNofGsuch thatHN=GandH∩N≤H_G, where[formula]is the maximal normal subgroup ofGwhich is contained inH. We obtain some results about thec-normal subgroups and use them to determine the structures of some groups.     

Conjugacy classes of maximal nilpotent subgroups

It is shown that in an arbitrary finite groupG, any two maximal nilpotent subgroups ofG whose intersection contains its own centralizer inG, are conjugate inG.     

Primitive permutation groups of simple diagonal type

LetG be a finite primitive group such that there is only one minimal normal subgroupM inG, thisM is nonabelian and nonsimple, and a maximal normal subgroup ofM is regular. Further, letH be a point stabilizer inG. ThenH∩M is a (nonabelian simple) common complement inM to all the maximal normal subgroups ofM, and there is a natural identification ofM with a direct powerT ^m of a nonabelian simple groupT in whichH∩M becomes the “diagonal” subgroup ofT ^m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic. GivenT ^m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal subgroups are abstractly isomorphic toT ^m. Secondly, form the direct productS _m×OutT of the symmetric group of degreem and the outer automorphism group ofT (so OutT=AutT/InnT), and consider the set of the conjugacy classes of those subgroups inS _m×OutT whose projections inS _m are primitive. The second result of the paper is that there is a bijection between these two sets. The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which can fall into a single abstract isomorphism class.     

Teoremi di normalizzazione per l'omotopia regolare dei grafi

Given a metric compact spaceS and a finite graphG we show that:

Linear core conditions in residually finite groups

LetG be a residually finite or pro-finite group. We say thatG satisfies the linear core condition with constantc if all finite index (open) subgroups ofG contain a subgroup of index at mostc which is normal inG. Answering a question of L. Pyber we give a complete characterisation of finitely generated residually finite and pro-finite groups satisfying a linear core generated residually finite and pro-finite groups satisfying a linear core condition. In the case of infinitely generated groups we prove that such groups are abelian-by-finite. Research supported by the Hungarian National Research Foundation (OTKA), grant no. 16432 and F023436.     

On c -Normal Maximal and Minimal Subgroups of Sylow Subgroups of Finite Groups. II

Abstract

A subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H _G  = Core(H). We extend the study on the structure of a finite group under the assumption that all maximal or minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of G are c-normal in G. The main theorems we proved in this paper are:

Theorem Let ? be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ?. If all maximal subgroups of any Sylow subgroup of F*(H) are c-normal in G, then G ∈ ?.

Theorem Let ? be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ?. If all minimal subgroups and all cyclic subgroups of F*(H) are c-normal in G, then G ∈ ?.     

Quasinormalität in topologischen Gruppen

LetQ be a subgroup of the locally compact groupG. Q is called a topologically quasinormal subgroup ofG, ifQ is closed and for each closed subgroupA ofG. We prove: If the compact elements ofG form a proper subgroup, compact topologically quasinormal subgroups ofG are subnormal of defect 2. IfG is connected, compact topologically quasinormal subgroups ofG are normal. IfG/G ₀ is compact, connected topologically quasinormal subgroups ofG are normal.     

Finite dimensional representations and subgroup actions on homogeneous spaces

LetH be an ℝ-subgroup of a ℚ-algebraic groupG. We study the connection between the dynamics of the subgroup action ofH onG/G _ℤ and the representation-theoretic properties ofH being observable and epimorphic inG. We show that ifH is a ℚ-subgroup thenH is observable inG if and only if a certainH orbit is closed inG/G _ℤ; that ifH is epimorphic inG then the action ofH onG/G _ℤ is minimal, and that the converse holds whenH is a ℚ-subgroup ofG; and that ifH is a ℚ-subgroup ofG then the closure of the orbit underH of the identity coset image inG/G _ℤ is the orbit of the same point under the observable envelope ofH inG. Thus in subgroup actions on homogeneous spaces, closures of ‘rational orbits’ (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.     

<Emphasis Type="Italic">G</Emphasis>-covering subgroup systems for the classes of supersoluble and nilpotent groups

LetF be a class of groups andG a group. We call a set Σ of subgroups ofG aG-covering subgroup system for the classF (or directly aF-covering subgroup system ofG) ifG ∈F whenever every subgroup in Σ is inF. In this paper, we provide some nontrivial sets of subgroups of a finite groupG which are simultaneouslyG-covering subgroup systems for the classes of supersoluble and nilpotent groups. Research of the first author is supported by the NNSF of China (Grant No. 10171086) and QLGCF of Jiangsu Province and a Croucher Fellowship of Hong Kong. Research of the second author is partially supported by a UGC (HK) grant #2060176 (2001/2002).     

Locally Finite Groups with All Subgroups Normal-by-(Finite Rank)

A group is said to have finite (special) rank ≤ sif all of its finitely generated subgroups can be generated byselements. LetGbe a locally finite group and suppose thatH/H_Ghas finite rank for all subgroupsHofG, whereH_Gdenotes the normal core ofHinG. We prove that thenGhas an abelian normal subgroup whose quotient is of finite rank (Theorem 5). If, in addition, there is a finite numberrbounding all of the ranks ofH/H_G, thenGhas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly (Theorem 4). These results are based on analogous theorems on locally finitep-groups, in which case the groupGis also abelian-by-finite (Theorems 2 and 3).     

On discrete subgroups containing a lattice in a horospherical subgroup

We showed in [Oh] that for a simple real Lie groupG with real rank at least 2, if a discrete subgroup Γ ofG contains lattices in two opposite horospherical subgroups, then Γ must be a non-uniform arithmetic lattice inG, under some additional assumptions on the horospherical subgroups. Somewhat surprisingly, a similar result is true even if we only assume that Γ contains a lattice in one horospherical subgroup, provided Γ is Zariski dense inG.     

Isomorphisms ofp-adic group rings

SupposeP is the ring ofp-adic integers,G is a finite group of orderp ⁿ , andPG is the group ring ofG overP. IfV _p (G) denotes the elements ofPG with coefficient sum one which are of order a power ofp, it is shown that the elements of any subgroupH ofV _p (G) are linearly independent overP, and if in additionH is of orderp ⁿ , thenPGPH. As a consequence, the lattice of normal subgroups ofG and the abelianization of the normal subgroups ofG are determined byPG.     

Modulus of continuity of harmonic functions

LetG be a bounded plane domain, the diameters of whose boundary components have a fixed positive lower bound. Letu be harmonic inG and continuous in the closureG ofG. Suppose that the modulus of continuity ofu on the boundary ofG is majorized by a function of a suitable type. We shall then obtain upper bounds for the modulus of continuity ofu inG. Further, we shall show that in some situations these estimates cannot be essentially improved. We shall also consider the same problem for certain bounded domains in space. Research partially supported by the U.S. National Science Foundation. AMS (1980) Classification. Primary 31A05.     

Some remarks on subgroups determined by certain ideals in integral group rings

LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andH∩K⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI ³(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I ²(G)I(H)+I(G)I(H)I(G)+I(H)I²(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH ²(G/H′, T)≤1, are computed. the subgroup ofG determined byI ⁿ(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained     

Amenable subgroups of semi-simple groups and proximal flows

We present a classification of maximal amenable subgroups of a semi-simple groupG. The result is that modulo a technical connectivity condition, there are precisely 2′ conjugacy classes of such subgroups ofG and we shall describe them explicitly. Herel is the split rank of the groupG. These groups are the isotropy groups of the action ofG on the Satake-Furstenberg compactification of the associated symmetric space and our results give necessary and sufficient conditions for a subgroup to have a fixed point in this compactification. We also study the action ofG on the set of all measures on its maximal boundary. One consequence of this is a proof that the algebraic hull of an amenable subgroup of a linear group is amenable. Supported in part by NSF Grant No. MPS-74-19876.     

Groups with infinitely many types of fixed subgroups

It is a theorem of Shor that ifG is a word-hyperbolic group, then up to isomrphism, only finitely many groups appear as fixed subgroups of automorphisms ofG. We give an example of a groupG acting freely and cocompactly on a CAT(0) square complex such that infinitely many non-isomorphic groups appear as fixed subgroups of automorphisms ofG. Consequently, Shor’s finiteness result does not hold if the negative curvature condition is relaxed to either biautomaticity or nonpositive curvature. D. T. Wise was supported by grants from FCAR and NSERC.     

p-Pseudomeasures and closed subgroups

LetPM _p (G) be the space of allp-pseudomeasures on a locally compact groupG. We show the existence of a conditional expectation fromPM _p (G) ontoPM _p (H) whereH is a closed normal subgroup ofG. As an application we give a new proof of the fact thatH is a set ofp-synthesis inG; we also get an inequality involving the operator norm of bounded measures onG. Moreover, in analogy with a theorem of Reiter, we obtain a result concerning the closed ideals of the Figà-Talamanca Herz algebra ofG.