Real groups transitive on complex flag manifolds

Let be a complex flag manifold. The compact real form of is transitive on . If is a noncompact real form, such transitivity is rare but occasionally happens. Here we work out a complete list of Lie subgroups of transitive on and pick out the cases that are noncompact real forms of .

     

On -supplemented maximal and minimal subgroups of Sylow subgroups of finite groups

This paper proves: Let be a saturated formation containing . Suppose that is a group with a normal subgroup such that .

(1) If all maximal subgroups of any Sylow subgroup of are -supple- mented in , then ;

(2) If all minimal subgroups and all cyclic subgroups with order 4 of are -supplemented in , then .

     

Dense subsets of maximally almost periodic groups

A (discrete) group is said to be maximally almost periodic if the points of are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology on a group is totally bounded if whenever there is such that . For purposes of this abstract, a family with a totally bounded topological group is a strongly extraresolvable family if (a)  \vert G\vert$">, (b) each is dense in , and (c) distinct satisfy ; a totally bounded topological group with such a family is a strongly extraresolvable topological group.

We give two theorems, the second generalizing the first.

Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup.

Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable.

Theorem 2. Let be maximally almost periodic. Then there are a subgroup of and a family such that

(i) is dense in every totally bounded group topology on ;

(ii) the family is a strongly extraresolvable family for every totally bounded group topology on such that ; and

(iii) admits a totally bounded group topology as in (ii).

Remark. In certain cases, for example when is Abelian, one must in Theorem 2 choose . In certain other cases, for example when the largest totally bounded group topology on is compact, the choice is impossible.

     

Finite-dimensional left ideals in some algebras associated with a locally compact group

Let be a locally compact group, let be its group algebra, let be its usual measure algebra, let be the second dual of with an Arens product, and let be the conjugate of the space of bounded, left uniformly continuous, complex-valued functions on with an Arens-type product. We find all the finite-dimensional left ideals of these algebras. We deduce that such ideals exist in and if and only if is compact, and in (except those generated by right annihilators of ) and if and only if is amenable.

     

groups and quasi-equivalence

A torsion-free abelian group is if every map from a pure subgroup of into lifts to an endomorphism of The class of groups has been extensively studied, resulting in a number of nice characterizations. We obtain some characterizations for the class of homogeneous groups, those homogeneous groups such that, for pure in every has a lifting to a quasi-endomorphism of An irreducible group is if and only if every pure subgroup of each of its strongly indecomposable quasi-summands is strongly indecomposable. A group is if and only if every endomorphism of is an integral multiple of an automorphism. A group has minimal test for quasi-equivalence ( if whenever and are quasi-isomorphic pure subgroups of then and are equivalent via a quasi-automorphism of For homogeneous groups, we show that in almost all cases the and properties coincide.

     

Types in reductive -adic groups: The Hecke algebra of a cover

In this paper, is a non-Archimedean local field and is the group of -points of a connected reductive algebraic group defined over . Also, is an irreducible representation of a compact open subgroup of , the pair being a type in . The pair is assumed to be a cover of a type in a Levi subgroup of . We give conditions, generalizing those of earlier work, under which the Hecke algebra is the tensor product of a canonical image of and a sub-algebra , for a compact open subgroup of containing .

     

Lifting wreath product extensions

Let and be finite groups and let be a hilbertian field. We show that if has a generic extension over and satisfies the arithmetic lifting property over , then the wreath product of and also satisfies the arithmetic lifting property over . Moreover, if the orders of and are relatively prime and is abelian, then any extension of by (which is necessarily a semidirect product) has the arithmetic lifting property.

     

Open subgroups of

Let be a locally compact group and let denote the -algebra generated by left translation operators on . Let and be the spaces of almost periodic and weakly almost periodic functionals on the Fourier algebra , respectively. It is shown that if contains an open abelian subgroup, then (1) if and only if is norm dense in ; (2) is a -algebra if is norm dense in , where denotes the set of elements in with compact support. In particular, for any amenable locally compact group which contains an open abelian subgroup, has the dual Bohr approximation property and is a -algebra.

     

Isomorphism of complete local noetherian rings and strong approximation

About a year ago Angus Macintyre raised the following question. Let and be complete local noetherian rings with maximal ideals and such that is isomorphic to for every . Does it follow that and are isomorphic? We show that the answer is yes if the residue field is algebraic over its prime field. The proof uses a strong approximation theorem of Pfister and Popescu, or rather a variant of it, which we obtain by a method due to Denef and Lipshitz. Examples by Gabber show that the answer is no in general.

     

Isomorphism of commutative group algebras of closed -local algebraically compact groups

Let be an abelian group and let be a field of 0$">. It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the -isomorphism for some group yields provided is a closed -group or a -local algebraically compact group. In particular, this is the case when is closed -primary of arbitrary power, or is -local algebraically compact with cardinality at most and is in cardinality not exceeding . The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).

     

Actions of pointed Hopf algebras with reduced pi invariants

Let be an -module algebra, where is a pointed Hopf algebra acting on finitely of dimension . Suppose that for every nonzero -stable left ideal of . It is proved that if satisfies a polynomial identity of degree , then satisfies a polynomial identity of degree provided at least one of the following additional conditions is fulfilled:

1. is semiprime and is almost central in ,
2. is reduced.

If we also assume that is central, then satisfies the standard polynomial identity of degree , where is the greatest integer in .

     

A compact group which is not Valdivia compact

A compact space is Valdivia compact if it can be embedded in a Tikhonov cube in such a way that the intersection is dense in , where is the sigma-product ( the set of points with countably many non-zero coordinates). We show that there exists a compact connected Abelian group of weight which is not Valdivia compact, and deduce that Valdivia compact spaces are not preserved by open maps.

     

A characterization of the projective line

Let be a set (with at least three different points) and let be a group of bijections of . If the action of on satisfies three natural conditions, then admits a canonical structure of a projective line over a commutative field, such that is the group of all projective transformations of .

     

Splitting sets in integral domains

Let be an integral domain. A saturated multiplicatively closed subset of is a splitting set if each nonzero may be written as where and for all . We show that if is a splitting set in , then is a splitting set in , a multiplicatively closed subset of , and that is a splitting set in is an lcm splitting set of , i.e., is a splitting set of with the further property that is principal for all and . Several new characterizations and applications of splitting sets are given.

     

Linearity of dimension functions for semilinear -spheres

In this paper, we show that the dimension function of every semilinear -sphere is equal to that of a linear -sphere for finite nilpotent groups of order , where , are primes. We also show that there exists a semilinear -sphere whose dimension function is not virtually linear for an arbitrary nonsolvable compact Lie group .

     

On a theorem of Barbara Schmid

Let be a finite group and a complex representation. Barbara Schmid has shown that the algebra of invariant polynomial functions on the vector space is generated by homogeneous polynomials of degree at most , where is the largest degree of a generator in a minimal generating set for , and is the complex regular representation of . In this note we give a new proof of this result, and at the same time extend it to fields whose characteristic is larger than , the order of the group .

     

A variant of the Reynolds operator

Let be a linearly reductive group over a field , and let be a -algebra with a rational action of . Given rational --modules and , we define for the induced -action on Hom a generalized Reynolds operator, which exists even if the action on Hom is not rational. Given an -module homomorphism , it produces, in a natural way, an -module homomorphism which is -equivariant. We use this generalized Reynolds operator to study properties of rational - modules. In particular, we prove that if is invariantly generated (i.e. ), then is a projective (resp. flat) -module provided that is a projective (resp. flat) -module. We also give a criterion whether an -projective (or -flat) rational --module is extended from an -module.

     

A pure subalgebra of a finitely generated algebra is finitely generated

We prove the following. Let be a Noetherian commutative ring, a finitely generated -algebra, and a pure -subalgebra of . Then is finitely generated over .

     

On Burgess's theorem and related problems

Let be a graph. We determine all graphs which are -like. We also prove that if are graphs, then in order that each -like continuum be -indecomposable for some it is necessary and sufficient that if is a graph, then is not -like for some integer with . This generalizes a well known theorem of Burgess.

     

On the completeness of factor rings

Let be a complete local domain containing the integers with maximal ideal such that is at least the cardinality of the real numbers. Let be a nonmaximal prime ideal of such that is a regular local ring. We construct an excellent local ring such that the completion of is , the generic formal fiber of is local with maximal ideal and if is a nonzero ideal of , then is complete.