Hall广义补和群的可解性

研究具有某些特殊性质的广义补,得到了一些可解性的判别条件.如果对G的任意Sylow p-子群P,p∈{2,3}∩丌(G),NG(P)在G中都存在广义补H使H/D是G/D的Hall子群且H/D为幂零群,其中D=(H∩ⅣG(P))G,那么G可解.     

Restrictions of Irreducible Representations of Classical Algebraic Groups to Root A 1-Subgroups

Abstract

Restrictions of irreducible representations of classical algebraic groups to root A ₁-subgroups, i.e., subgroups of type A ₁ generated by root subgroups associated with two opposite roots, are studied. Composition factors of such restrictions are found in the following cases: for groups of types A _n with n > 2 and D _n , for groups of type B _n , n > 2, and long root subgroups, for groups of type C _n , n > 2, and short root subgroups, and for p-restricted representations of A ₂(K), C ₂(K) (recall that B ₂(K) ? C ₂(K)), and of B _n (K), n > 2, and short root subgroups. Here we assume that p > 2 for G = B _n (K) or C _n (K).     

Unique Node Extremal Subgroups in Chevalley Groups

Abstract

Taking G to be a Chevalley group of rank at least 3 and U to be the unipotent radical of a Borel subgroup B,an extremal subgroup A is an abelian normal subgroup of U which is not contained in the intersection of all the unipotent radicals of the rank 1 parabolic subgroups of G containing B. If there is an unique rank 1 parabolic subgroup P of G containing B with the property that A is not contained in the unipotent radical of P,then A is called a unique node extremal subgroup. In this paper we investigate the embedding of unique node extremal subgroups in U and prove that,apart from some specified cases,such a subgroup is contained in the unipotent radical of a certain maximal parabolic subgroup.     

weyl群的定义关系及其应用

首 先深化 Ｒ． Ｓｔｅｉｎｂｅｒｇ 关于 Ｗ ｅｙｌ 群定义 关系的一个定 理,作为应 用,对 Ｂｌ , Ｃｌ 型 Ｗ ｅｙｌ群分别构造了一 个指数为２的正 规子群     

On the normality of the rings of Schubert varieties

We prove that the cone over a Schubert variety inG/P (P being a maximal parabolic subgroup of classical type) is normal by exhibiting a 2-regular sequence inR(w) (the homogeneous coordinate ring of the Schubert varietyX(w) inG/P under the canonical protective embeddingG/P ⊂→ (p (H° G/P,L)),L being the ample generator of (PicG/P), which vanishes on the singular locus ofX(w). We also prove the surjectivity ofH° (G/Q, L) H° (X(w), L), whereQ is a classical parabolic subgroup (not necessarily maximal) ofG andL is an ample line bundle onG/Q.     

Special-valued subgroups of lattice-ordered groups

We prove that the intersection of all maximal special-valued subgroups of a lattice-ordered group is the special-valued quasi-torsion radical of a lattice-ordered group , which extends our earlier result that the intersection of all maximal finite-valued subgroups of a lattice-ordered group is the finite-valued torsion radical of . We also show that the class of almost finite-valued lattice-ordered groups is a quasi-torsion class, and the quasi-torsion radical of a group is equal to the intersection of the group with the lateral completion of the finite-valued torsion radical of the group.

     

Algebraic Lie algebras with complete Borel subalgebras

In this article we prove that an algebraic Lie algebra over an algebraically closed field of characteristic 0 is complete if its Borel subalgebras are complete. Thus the study on complete Lie algebras may somewhat be reduced to that on solvable complete ones.     

Finite <Emphasis Type="Italic">p</Emphasis>-groups with a class of complemented normal subgroups

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.     

Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X ₀(N) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X ₀(N), if N=pM with p prime, p ∤ M and the genus of X ₀(M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin–Lehner involution. This gives, in a sense, a generalization of Ogg’s Theorem in some cases.