Unrefined minimal K-types forp-adic groups

Oblatum 23-II-1993 & 18-V-1993     

A binomial product representation forp-adic numbers

Archiv der Mathematik -     

A solvability criterion for finite groups

We show that if for every prime p, the normalizer of a Sylow p-subgroup of a finite group G admits a p-solvable supplement, then G is solvable. This generalizes a solvability criterion of Hall which asserts that a finite group G is solvable if and only if G has a Hall p′-subgroup for every prime p.     

A solubility criterion for finite groups

It is proved that a finite group is soluble if the normalizer of every cyclic subgroup of prime-power order has a soluble supplement.Received: 31 January 2005     

A criterion for the p-solubility of finite groups

Finite groups, whose p-complements commute with a definite set of subgroups of a Sylow p-subgroup, are investigated. It is proved that such groups are p-soluble and have a p-length equal to 1.Translated from Matematicheskie Zametki, Vol. 9, No. 4, pp. 375–383, April, 1971.     

A conjugacy criterion for Hall subgroups in finite groups

A finite group G is said to satisfy C _π for a set of primes π, if G possesses exactly one class of conjugate π-Hall subgroups. We obtain a criterion for a finite group G to satisfy C _π in terms of a normal series of the group.     

A criterion for subnormality of subgroups in finite groups

Based on Wielandt’s criterion for subnormality of subgroups in finite groups, we study 2-maximal subgroups of finite groups and present another subnormality criterion in finite solvable groups.     

A local criterion for Weyl modules for groups of type A

Let G be a universal Chevalley group over an algebraically closed field and U⁻ be the subalgebra of generated by all divided powers X_α,m with α<0. We conjecture an algorithm to determine if , where F∈U⁻, ω is a dominant weight and is a highest weight vector of the Weyl module Δ(ω). This algorithm does not use bases of Δ(ω) and is similar to the algorithm for irreducible modules that involves stepwise raising the vector under investigation. For an arbitrary G, this conjecture is proved in one direction and for G of type A in both.     

Distal criterion of transitive transformation groups

Let G be a topological group, H a closed subgroup of the group G, and G/H a homogeneous space of cosets Hg(g G). The group G acts naturally on G/H, defining a transitive transformation group (G/H, G,gp), (Ha, g)=Hag(a G, g G). Necessary and sufficient conditions for the distalness of the transformation group (G/H, G,) are indicated.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 77–80, January, 1969.