A group is subdirectly reducible if it has two non-trivial normal subgroups with trivial intersection. Such groups may be an easy case in certain inductive arguments. We prove that every solvable finite group can be generated by at most one subdirectly reducible subgroup together with two subgroups of prime-power order and one element. We also prove that every group of prime-power order can be ...

Let p be a prime number, G a finite group, Q a p-subgroup of G and L any subgroup of G containing the normalizer NG(Q) of Q in G. Let k be a field of positive characteristic p. In [2] J.A. Green demonstrates a fundamental correspondence between finitely generated kG-modules with vertex Q and finitely generated kL-modules with vertex Q. When L = NG(Q) the Green correspondence allows for the redu...

The Schubert cells eλ are the orbits of the Iwahori subgroup B̃, while the Birkhoff strata Sλ are the orbits of the opposite Iwahori subgroup B̃ . The cells and the strata are dual in the sense that Sλ ∩ eλ = {λ}, and the intersection is transverse. The closure of eλ is the affine Schubert variety Xλ. It has dimension l (λ), where l is the minimal length occuring in the coset λW̃I , and its cells ...

where a, b, c, d are rational integers and ad-bc= 1. Then T is generated by the linear fractional transformations S, T where St = t+ 1, 7V= — 1/t and is the free product of the cyclic group {T} of order 2 and the cyclic group {ST} of order 3. Let G be a normal subgroup of T of finite index p. It is known that there is just one such subgroup for p=l, 2, 3 which may be described as T", the subgro...

let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal h $ -subgroup in‎ ‎$g$ if $n_g (h)cap h^gleq h$ for all $gin‎ ‎g$. a subgroup $h$ of $g$ is called a weakly‎ $mathcal h^ast $-subgroup in $g$ if there exists a‎ ‎subgroup $k$ of $g$ such that $g=hk$ and $hcap‎ ‎k$ is an $mathcal h$-subgroup in $g$. we‎ ‎investigate the structure of the finite group $g$ under the‎ ‎assump...

Abstract We introduce the concept of alternate-edge-colourings for maps and study highly symmetric examples such maps. Edge-biregular type ( k , l ) occur as smooth normal quotients a particular index two subgroup $$T_{k,l}$$ T k , l </m...

The cogrowth of a subgroup is defined as the growth of a set of coset representatives which are of minimal length. A subgroup is essential if it intersects non-trivially every non-trivial subgroup. The main result of this paper is that every function f : N ∪ {0}−→N which is strictly increasing, but at most exponential, is equivalent to a cogrowth function of an essential subgroup of infinite in...

In this paper we prove a generalization of conjectures of Bogomolov and Lang in terms of an arithmetic Néron-Tate height pairing over a finitely generated field. §0. Introduction Let K be a finitely generated field over Q with d = tr. degQ(K), and let B be a big polarization of K . Let A be an abelian variety over K , and let L be a symmetric ample line bundle on A. In [4] we define the height ...

It is proved here that each finite completely reducible linear group of dimension d (over an arbitrary field) can be generated by L 3 d] elements. If a finite linear group G of dimension d is not completely reducible, then its characteristic is a prime, p say, and the factor group of G modulo the largest normal p-subgroup O.p(G) may be viewed as a completely reducible linear group acting on the...

Let G be a connected semisimple Lie group with finite center and without compact factor, and Γ a lattice in G, that is, a discrete subgroup of G such that Γ\G has finite volume. In this article we investigate the distribution of orbits of Γ acting on the Furstenberg boundary of G. Recall that the Furstenberg boundary can be identified with the factor space G/P , where P is a minimal parabolic s...