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Yoav Segev - One of the best experts on this subject based on the ideXlab platform.
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finite quotients of the Multiplicative Group of a finite dimensional division algebra are solvable
Journal of the American Mathematical Society, 2002Co-Authors: Andrei S Rapinchuk, Yoav Segev, Gary M SeitzAbstract:We prove that finite quotients of the Multiplicative Group of a finite dimensional division algebra are solvable. Let D be a finite dimensional division algebra having center K and let N ⊆ D× be a normal subGroup of finite index. Suppose D×/N is not solvable. Then we may assume that H := D×/N is a minimal nonsolvable Group (MNS Group for short), i.e., a nonsolvable Group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS Group satisfies a certain property which we denote property (3 12 ). This property includes the requirement that the diameter of the commuting graph should be ≥ 3, but is, in fact, stronger. Another ingredient is to show that if the commuting graph of D×/N has the property (312 ), then N is open with respect to a nontrivial height one valuation of D (assuming without loss, as we may, that K is finitely generated). After establishing the openness of N (when D×/N is an MNS Group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of K over its prime subfield, to eliminate H as a possible quotient of D×, thereby obtaining a contradiction and proving our main result.
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the finite quotients of the Multiplicative Group of a division algebra of degree 3 are solvable
Israel Journal of Mathematics, 1999Co-Authors: Louis Rowen, Yoav SegevAbstract:LetD be a finite dimensional division algebra. It is known that in a variety of cases, questions about the normal subGroup structure ofDx (the Multiplicative Group ofD) can be reduced to questions about finite quotients ofDx. In this paper we prove that when deg(D)=3, finite quotients ofDx are solvable. the proof uses Wedderburn’s Factorization Theorem.
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on finite homomorphic images of the Multiplicative Group of a division algebra
arXiv: Rings and Algebras, 1999Co-Authors: Yoav SegevAbstract:This paper, together with a forthcoming paper by the author and Seitz, proves the Margulis-Platonov conjecture concerning the normal subGroup structure of algebraic Groups over number fields, in the case of inner forms of anisotropic Groups of type $A_n$.
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on finite homomorphic images of the Multiplicative Group of a division algebra
Annals of Mathematics, 1999Co-Authors: Yoav SegevAbstract:Conjecture 2 (G. Margulis and V. Platonov). Let G be a simple, simply connected algebraic Group defined over an algebraic number field K. Let T be the set of all nonarchimedean places v of K such that G is Kv-anisotropic; then for any noncentral normal subGroup N ≤ G(K) there exists an open normal subGroup W ≤ G(K,T ) = ∏ v∈T G(Kv) such that N = G(K) ∩ W ; in particular, if T = ∅ then G(K) does not have proper noncentral normal subGroups.
Jairo Z Goncalves - One of the best experts on this subject based on the ideXlab platform.
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free Groups in a normal subGroup of the field of fractions of a skew polynomial ring
Communications in Algebra, 2017Co-Authors: Jairo Z GoncalvesAbstract:ABSTRACTLet k(t) be the field of rational functions over the field k, let σ be a k-automorphism of K = k(t), let D = K(X;σ) be the ring of fractions of the skew polynomial ring K[X;σ], and let D∙ be the Multiplicative Group of D. We show that if N is a noncentral normal subGroup of D∙, then N contains a free subGroup. We also prove that when k is algebraically closed and σ has infinite order, there exists a specialization from D to a quaternion algebra. This allows us to explicitly present free subGroups in D∙.
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free symmetric and unitary pairs in division rings infinite dimensional over their centers
Israel Journal of Mathematics, 2015Co-Authors: Vitor O. Ferreira, Jairo Z GoncalvesAbstract:Let D be a division ring infinite-dimensional over its center k with Multiplicative Group D ×. We show that if D belongs to certain families, there exist free symmetric and unitary pairs in D × with respect to a k-involution on D.
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free Groups in normal subGroups of the Multiplicative Group of a division ring
Journal of Algebra, 2015Co-Authors: Jairo Z Goncalves, D S PassmanAbstract:Abstract Let D be a division ring with center Z and Multiplicative Group D ∖ { 0 } = D • , and let N be a normal subGroup of D • . We investigate various conditions under which N must contain a free noncyclic subGroup. In one instance, assuming that the transcendence degree of Z over its prime field is infinite, and that N contains a nonabelian solvable subGroup, we use a construction method due to Chiba to exhibit free generators of the free subGroup.
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free symmetric and unitary pairs in division rings infinite dimensional over their centers
arXiv: Rings and Algebras, 2013Co-Authors: Vitor O. Ferreira, Jairo Z GoncalvesAbstract:Let $D$ be a division ring infinite-dimensional over its center $k$ with Multiplicative Group $D^{\times}$. We show that if $D$ belongs to certain families, there exist free symmetric and unitary pairs in $D^{\times}$ with respect to a $k$-involution on $D$.
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A Survey on Free Objects in Division Rings and in Division Rings with an Involution
Communications in Algebra, 2012Co-Authors: Jairo Z Goncalves, M. ShirvaniAbstract:Let D be a division ring with center k, and let D † be its Multiplicative Group. We investigate the existence of free Groups in D †, and free algebras and free Group algebras in D. We also go through the case when D has an involution * and consider the existence of free symmetric and unitary pairs in D †.
Sophie Huczynska - One of the best experts on this subject based on the ideXlab platform.
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The strong primitive normal basis theorem
Acta Arithmetica, 2010Co-Authors: Stephen D. Cohen, Sophie HuczynskaAbstract:An element α of the extension E of degree n over the finite field F = GF(q) is called free over F if {α, α q , . . . , α q n 1 } is a (normal) basis of E/F. The primitive normal basis theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F, there exists an element α ∈ E such that α is simultaneously primitive (i.e., generates the Multiplicative Group of E) and free over F. In this paper we prove the following strengthening of this theorem: aside from four specific extensions E/F, there exists an element α ∈ E such that both α and α −1 are simultaneously primitive and free over F. AMS classification: Primary 11T30; Secondary 11T06, 12E20
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The Strong Primitive Normal Basis Theorem
arXiv: Number Theory, 2006Co-Authors: Stephen D. Cohen, Sophie HuczynskaAbstract:An element w of the extension E of degree n over the finite field F=GF(q) is called free over F if {w, w^q,...,w^{q^{n-1}}} is a (normal) basis of E/F. The Primitive Normal Basis Theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F, there exists an element w in E such that w is simultaneously primitive (i.e., generates the Multiplicative Group of E) and free over F. In this paper we prove the following strengthening of this theorem: aside from five specific extensions E/F, there exists an element w in E such that both w and w^{-1} are simultaneously primitive and free over F.
Greg Martin - One of the best experts on this subject based on the ideXlab platform.
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the smallest invariant factor of the Multiplicative Group
International Journal of Number Theory, 2020Co-Authors: Ben Chang, Greg MartinAbstract:Let λ1(n) denote the least invariant factor in the invariant factor decomposition of the Multiplicative Group Mn = (ℤ/nℤ)×. We give an asymptotic formula, with order of magnitude x/log x, for the c...
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the distribution of the number of subGroups of the Multiplicative Group
Journal of The Australian Mathematical Society, 2020Co-Authors: Greg Martin, Lee TroupeAbstract:Let $I(n)$ denote the number of isomorphism classes of subGroups of $(\Bbb Z/n\Bbb Z)^\times$, and let $G(n)$ denote the number of subGroups of $(\Bbb Z/n\Bbb Z)^\times$ counted as sets (not up to isomorphism). We prove that both $\log G(n)$ and $\log I(n)$ satisfy Erdos-Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that $\log G(n)$ is not an additive function but is closely related to the sum of squares of additive functions. We also establish the orders of magnitude of the maximal orders of $\log G(n)$ and $\log I(n)$.
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the smallest invariant factor of the Multiplicative Group
arXiv: Number Theory, 2019Co-Authors: Ben Chang, Greg MartinAbstract:Let $\lambda_1(n)$ denote the least invariant factor in the invariant factor decomposition of the Multiplicative Group $M_n = (\mathbb Z/n\mathbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/\sqrt{\log x}$, for the counting function of those integers $n$ for which $\lambda_1(n)\ne2$. We also give an asymptotic formula, for any even $q\ge4$, for the counting function of those integers $n$ for which $\lambda_1(n)=q$. These results require a version of the Selberg-Delange method whose dependence on certain parameters is made explicit, which we provide in an appendix. As an application, we give an asymptotic formula for the counting function of those integers $n$ all of whose prime factors lie in an arbitrary fixed set of reduced residue classes, with implicit constants uniform over all moduli and sets of residue classes.
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uniform bounds for the least almost prime primitive root
arXiv: Number Theory, 1998Co-Authors: Greg MartinAbstract:We investigate, using the weighted linear sieve, the distribution of almost-primes among the residue classes (mod p) that generate the Multiplicative Group of reduced residue classes. We are concerned with finding an upper bound for the least prime or almost-prime primitive root (mod p) that holds uniformly for all p, analogous to Linnik's Theorem on a uniform upper bound for the least prime in a single arithmetic progression (mod p).
Wolfgang M Schmidt - One of the best experts on this subject based on the ideXlab platform.
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linear equations in variables which lie in a Multiplicative Group
arXiv: Number Theory, 2004Co-Authors: Janhendrik Evertse, Hans Peter Schlickewei, Wolfgang M SchmidtAbstract:Let K be a field of characteristic 0 and let n be a natural number. Let Gamma be a subGroup of the Multiplicative Group $(K^\ast)^n$ of finite rank r. Given $A_2,...,a_n\in K^\ast$ write $A(a_1,...,a_n,\Gamma)$ for the number of solutions x=(x_1,...,x_n)\in \Gamma$ of the equation a_1x_1+...+a_nx_n=1$, such that no proper subsum of $a_1x_1+...+a_nx_n$ vanishes. We derive an explicit upper bound for $A(a_1,...,a_n,\Gamma)$ which depends only on the dimension n and on the rank r.
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linear equations in variables which lie in a Multiplicative Group
Annals of Mathematics, 2002Co-Authors: Janhendrik Evertse, Hans Peter Schlickewei, Wolfgang M SchmidtAbstract:Let K be a field of characteristic 0 and let n be a natural number. Let r be a subGroup of the Multiplicative Group (K*) n of finite rank r. Given a 1 ,...,a n E K* write A(a 1 ,...,a n , Γ) for the number of solutions x = (x 1 ,..., x n ) ∈ Γ of the equation a 1 x 1 +... + a n x n = 1, such that no proper subsum of a 1 x 1 + ...+ a n x n vanishes. We derive an explicit upper bound for A(a 1 ,.., a n , r) which depends only on the dimension n and on the rank r.
