Locally compact subgroups of the spectrum of the measure algebra II

The maximal ideal space Δ_G of the measure algebra M(G) of a locally compact abelian group G is a compact commutative semitopological semigroup. In this paper we show that cℓ Ĝ the closure of Ĝ, the dual of G, in Δ_G can contain maximal subgroups which are not locally compact. We have previously characterized the locally compact maximal subgroups of cℓ Ĝ as arising from locally compact topologies on G which are finer than the original topology. This research was supported in part by NSF contract number GP-19852.     

Normalizers of subsystem subgroups in finite groups of Lie type

Finite groups of Lie type form the greater part of known finite simple groups. An important class of subgroups of finite groups of Lie type are so-called reductive subgroups of maximal rank. These arise naturally as Levi factors of parabolic groups and as centralizers of semisimple elements, and also as subgroups with maximal tori. Moreover, reductive groups of maximal rank play an important part in inductive studies of subgroup structure of finite groups of Lie type. Yet a number of vital questions dealing in the internal structure of such subgroups are still not settled. In particular, we know which quasisimple groups may appear as central multipliers in the semisimple part of any reductive group of maximal rank, but we do not know how normalizers of those quasisimple groups are structured. The present paper is devoted to tackling this problem. Supported by RFBR (grant No. 05-01-00797) and by SB RAS (Young Researchers Support grant No. 29 and Integration project No. 2006.1.2). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 3–30, January–February, 2008.     

Finite non-cyclic p -groups whose number of subgroups is minimal

Recent results of Qu and Tărnăuceanu explicitly describe the finite p-groups which are not elementary Abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results from the bottom level up by completely determining the non-cyclic finite p-groups whose number of subgroups among p-groups of a given order is minimal.     

On fixed subgroups of maximal rank

We show that, in the free group F of rank n, n is the maximal length of strictly ascending chains of maximal rank fixed subgroups, that is, rank n subgroups of the form Fix^{^} for some 4> L Aut(F). We further show that, in the rank two case, if the intersection of an arbitrary family of proper maximal rank fixed subgroups has rank two then all those subgroups are equal. In particular, every G < Aut(F) with r(FixG) = 2 is either trivial or infinite cyclic.     

Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable

Thompson’s theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson’s theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.     

On s-quasinormal and c-normal subgroups of a finite group

Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown： （1） G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; （2） G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F（H）, the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G.     

Groups with maximal subgroups of Sylow subgroups normal

This paper characterizes those finite groups with the property that maximal subgroups of Sylow subgroups are normal. They are all certain extensions of nilpotent groups by cyclic groups.     

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.     

A note on s-conditional permutability of maximal subgroups of Sylow subgroups of finite groups

Monatshefte für Mathematik - In this note, we point out an error in the paper “On s-conditional permutability of maximal subgroups of Sylow subgroups of finite groups” (Monatsh...     

Dual-standard subgroups of finite and locally finite groups

A subgroup D of a group G is called dual-standard if, for all subgroups X and Y of G, <XχD,YχD>=<X,Y>χD. When G is finite, Zappa has given some information concerning the way in which D is embedded in G and the structure of G itself. Among other things, Zappa makes reference to the maximal normal Hall subgroup L of G contained in D. In general L can be arbitrary. The main result of this work, however, is to show that the commutator of L with a complement in G is nilpotent.     

On finite groups with p-decomposable cofactors of subgroups

It is proved that, for a finite group with p-decomposable cofactors of the maximal subgroups, the quotient group by the Fitting subgroup is p-decomposable. This implies that every group all of whose maximal subgroups have nilpotent cofactors is metanilpotent.     

On second maximal subgroups of Sylow subgroups of finite groups

Finite groups in which the second maximal subgroups of the Sylow p-subgroups, p a fixed prime, cover or avoid the chief factors of some of its chief series are completely classified.     

Explicit representations of maximal subgroups of the Monster

Most of the maximal subgroups of the Monster are now known, but in many cases they are hard to calculate in. We produce explicit ‘small’ representations of all the maximal subgroups which are not 2-local. The representations we construct are available on the World Wide Web at http://brauer.maths.qmul.ac.uk/Atlas/.     

Normal subgroups of finite groups and formations with normalizer conditions

The structure of a normal subgroupK of a finite groupG is studied under the condition that nontrivial intersections ofK with maximal subgroups ofG belong to an arbitrary formation with normalizer condition. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 867–870, December, 1999.     

Frattini subgroups and supernilpotent groups

In the context of the problem of which nonabelianp-groups can occur as normal subgroups contained in Frattini subgroups, the family of supernilpotent groups (all maximal subgroups characteristic) is investigated. Results of this investigation are applied to the Frattini-embedding problem, incorporating recent work of A. R. Makan. The groups of order 2ⁿ (n ≦ 6) have been examined with respect to supernilpotence and their occurrence as normal subgroups contained in Frattini subgroups. Results of this examination are presented.     

次正规子群与有限群的超可解性

通过Sylow子群的极大子群和次正规性,利用极小阶反例的方法,得出群p-幂零性和超可解性的结论.本文的创新改进之处在于结合Sylow子群的极大子群和次正规性,研究p-幂零性和超可解性的相关结论.     

On the intersection of maximal subgroups in finite groups

The intersections of maximal subgroups of a definite kind in finite groups are investigated. In particular, it is proved that the intersection of all noninvariant nonnilpotent maximal subgroups is nilpotent in a finite nonsolvable group.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 429–439, September, 1973.     

On the normal indices of proper subgroups of finite groups

We extended the normal index from maximal subgroups to proper subgroups. We give a quantitative version of all results obtained by using c-normal subgroups and obtain some new characterizations of solvable, supersolvable and nilpotent groups by the normal indices of proper subgroups.     

Compact lie group actions with isotropy subgroups of maximal rank

When a compact connected Lie group acts smoothly on a manifold X with only connected isotropy subgroups of maximal rank, the action is completeley determined by the corresponding action of its Weylgroup WG on the fixed space X^T of the maximal torus. Isotropy subgroups of such actions are determined.     

Congruence subgroups of Hecke groups

Hecke groups are an important tool in subgroups of Hecke groups play an important rule investigating functional equations, and congruence in research of the solutions of the Dirichlet series. When q, m are two primes, congruence subgroups and the principal congruence subgroups of level m of the Hecke group H（√q） have been investigated in many papers. In this paper, we generalize these results to the case where q is a positive integer with q ≥ 5, √q ￠ Z and m is a power of an odd prime.