We study finite groups G with the property that for any subgroup M maximal in whose order is divisible by all prime divisors of |G|, supersolvable. show nonabelian simple group can occur as a composition factor such and that, if solvable, then nilpotency length rank are arbitrarily large. On other hand, every p, p-length at most 1. This answers questions proposed V. Monakhov The Kourovka Notebo...

Let k be an algebraic number field. Let α be a root of a polynomial f ∈ k[x] which is solvable by radicals. Let L be the splitting field of α over k. Let n be a natural number divisible by the discriminant of the maximal abelian subextension of L, as well as the exponent of G(L/k), the Galois group of L over k. We show that an optimal nested radical with roots of unity for α can be effectively ...

Abstract It is an interesting and difficult topic to determine the structure of a finite group by number elements maximal order. This related Thompson’s conjecture, that is, if two groups have same order type one them solvable, then other solvable. In this article, we continue work prove G G which has 4 <m...

We reduce a case of the hidden subgroup problem (HSP) in several families of finite groups of Lie type, such as SL(2; q), PSL(2; q), and PGL(2; q), to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective space in an “almost” 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup. Our observation ...

Let $X$ be a surface, possibly with boundary. Suppose it has infinite genus or infinitely many punctures, closed subset which is disk Cantor set removed from its interior. For example, could any surface of type only finitely boundary components. We prove that the mapping class group does not satisfy Tits Alternative. That is, Map$(X)$ contains generated subgroup virtually solvable and no nonabe...

Let Γ be a discrete subgroup of a simply connected, solvable Lie group G, such that AdG Γ has the same Zariski closure as AdG. If α : Γ → GLn(R) is any finite-dimensional representation of Γ, we show that α virtually extends to a continuous representation σ of G. Furthermore, the image of σ is contained in the Zariski closure of the image of α. When Γ is not discrete, the same conclusions are t...

Let X be a compact Kähler manifold of complex dimension n, and let G be a connected solvable subgroup of the automorphism group Aut(X). Let N(G) be the normal subgroup of G of elements of null entropy. The aim of this short paper is to show that G/N(G) is a free abelian group of rank ≤ n− 1 and that the rank estimate is optimal. This proves the conjecture of Tits type in [6] affirmatively. The ...

We define the toplogical generating rank dtop(G) of a topological group G as the minimal number of elements of G which generate a dense subgroup of G. I will present joint work with Gena Noskov, partly in progress, about our attempts to compute the topological generating rank for connected Lie groups. The cases that G is semisimple, where dtop(G) is slightly bigger than one, and that G is nilpo...