Logarithmic diameter bounds for some Cayley graphs

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Lam Pham ◽  
Xin Zhang
Keyword(s):  
Positive Integer ◽  
Cayley Graph ◽  
Linear Group ◽  
Canonical Projection ◽  
Zariski Closure ◽  
Linear Groups ◽  
Arbitrary Positive Integer ◽  
Special Linear Groups ◽  
Reduction Modulo ◽  
Symmetric Set

Abstract Let S ⊂ GL n ⁢ ( Z ) S\subset\mathrm{GL}_{n}(\mathbb{Z}) be a finite symmetric set. We show that if the Zariski closure of Γ = ⟨ S ⟩ \Gamma=\langle S\rangle is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph Cay ⁡ ( Γ / Γ ⁢ ( q ) , π q ⁢ ( S ) ) \operatorname{Cay}(\Gamma/\Gamma(q),\pi_{q}(S)) is O ⁢ ( log ⁡ q ) O(\log q) , where 𝑞 is an arbitrary positive integer, π q : Γ → Γ / Γ ⁢ ( q ) \pi_{q}\colon\Gamma\to\Gamma/\Gamma(q) is the canonical projection induced by the reduction modulo 𝑞, and the implied constant depends only on 𝑆.

scholarly journals Right weak Engel conditions on linear groups

2019 ◽  
Vol 114 (1) ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Positive Integer ◽  
Linear Group ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Linear Groups ◽  
Locally Finite ◽  
Class A ◽  
Special Cases ◽  
Prüfer Rank

AbstractIf X is a group-class, a group G is right X-Engel if for all g in G there exists an X-subgroup E of G such that for all x in G there is a positive integer m(x) with [g, nx] ∈ E for all n ≥ m(x). Let G be a linear group. Special cases of our main theorem are the following. If X is the class of all Chernikov groups, or all finite groups, or all locally finite groups, then G is right X-Engel if and only if G has a normal X-subgroup modulo which G is hypercentral. The same conclusion holds if G has positive characteristic and X is one of the following classes; all polycyclic-by-finite groups, all groups of finite Prüfer rank, all minimax groups, all groups with finite Hirsch number, all soluble-by-finite groups with finite abelian total rank. In general the characteristic zero case is more complex.

scholarly journals Principal indecomposable modules for some three-dimensional special linear groups

1980 ◽  
Vol 22 (3) ◽  
pp. 439-455 ◽  
Author(s):  
James Archer
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
Three Dimensional ◽  
Special Linear Group ◽  
Linear Groups ◽  
Indecomposable Modules ◽  
Special Linear ◽  
Special Linear Groups ◽  
Irreducible Modules ◽  
Characteristic 2

Let k be a finite field of characteristic 2, and let G be the three dimensional special linear group over k. The principal indecomposable modules of G over k are constructed from tensor products of the irreducible modules, and formulae for their dimensions are given.

scholarly journals ON IMAGES OF REAL REPRESENTATIONS OF SPECIAL LINEAR GROUPS OVER COMPLETE DISCRETE VALUATION RINGS

2015 ◽  
Vol 58 (1) ◽  
pp. 263-272
Author(s):  
TALIA FERNÓS ◽  
POOJA SINGLA
Keyword(s):  
Linear Group ◽  
Positive Characteristic ◽  
Residue Field ◽  
Linear Groups ◽  
Special Linear ◽  
Special Linear Groups ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Abstract Homomorphisms ◽  
Finite Index Subgroup

AbstractIn this paper, we investigate the abstract homomorphisms of the special linear group SLn($\mathfrak{O}$) over complete discrete valuation rings with finite residue field into the general linear group GLm($\mathbb{R}$) over the field of real numbers. We show that for m < 2n, every such homomorphism factors through a finite index subgroup of SLn($\mathfrak{O}$). For $\mathfrak{O}$ with positive characteristic, this result holds for all m ∈ ${\mathbb N}$.

Hit problem for the polynomial algebra as a module over the Steenrod algebra in some degrees

2021 ◽  
pp. 2250007
Author(s):  
Nguyễn Khắc Tín
Keyword(s):  
Positive Integer ◽  
Linear Group ◽  
General Linear Group ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Arbitrary Positive Integer ◽  
Minimal Set ◽  
Prime Field ◽  
Set Up ◽  
Hit Problem

Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables with the degree of each [Formula: see text] being [Formula: see text] regarded as a module over the mod-[Formula: see text] Steenrod algebra [Formula: see text] and let [Formula: see text] be the general linear group over the prime field [Formula: see text] which acts naturally on [Formula: see text]. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra [Formula: see text] as a module over the mod-2 Steenrod algebra, [Formula: see text]. These results are used to study the Singer algebraic transfer which is a homomorphism from the homology of the mod-[Formula: see text] Steenrod algebra, [Formula: see text] to the subspace of [Formula: see text] consisting of all the [Formula: see text]-invariant classes of degree [Formula: see text] In this paper, we explicitly compute the hit problem for [Formula: see text] and the degree [Formula: see text] with [Formula: see text] an arbitrary positive integer. Using this result, we show that Singer’s conjecture for the algebraic transfer is true in the case [Formula: see text] and the above degree.

Trace polynomials of words in special linear groups

1979 ◽  
Vol 28 (4) ◽  
pp. 401-412 ◽  
Author(s):  
J. B. Southcott
Keyword(s):  
Linear Group ◽  
Characteristic Zero ◽  
Special Linear Group ◽  
Linear Groups ◽  
Two Dimensional ◽  
Arbitrary Mapping ◽  
Special Linear ◽  
Special Linear Groups ◽  
Integer Coefficients

AbstractIf w is a group word in n variables, x1,…,xn, then R. Horowitz has proved that under an arbitrary mapping of these variables into a two-dimensional special linear group, the trace of the image of w can be expressed as a polynomial with integer coefficients in traces of the images of 2n−1 products of the form xσ1xσ2…xσm 1 ≤ σ1 < σ2 <… <σm ≤ n. A refinement of this result is proved which shows that such trace polynomials fall into 2n classes corresponding to a division of n-variable words into 2n classes. There is also a discussion of conditions which two words must satisfy if their images have the same trace for any mapping of their variables into a two-dimensional special linear group over a ring of characteristic zero.

scholarly journals Holomorphic Cohomology of Complex Analytic Linear Groups

1966 ◽  
Vol 27 (2) ◽  
pp. 531-542 ◽  
Author(s):  
G. Hochschild ◽  
G. D. Mostow
Keyword(s):  
Linear Group ◽  
Structure Theory ◽  
Linear Groups ◽  
Dimensional Representation ◽  
Lie Algebra Cohomology ◽  
Analytic Group ◽  
Finite Dimensional Representation ◽  
Semidirect Products ◽  
Finite Dimensional ◽  
Rational Cohomology

Let G be a complex analytic group, and let A be the representation space of a finite-dimensional complex analytic representation of G. We consider the cohomology for G in A, such as would be obtained in the usual way from the complex of holomorphic cochains for G in A. Actually, we shall use a more conceptual categorical definition, which is equivalent to the explicit one by cochains. In the context of finite-dimensional representation theory, nothing substantial is lost by assuming that G is a linear group. Under this assumption, it is the main purpose of this paper to relate the holomorphic cohomology of G to Lie algebra cohomology, and to the rational cohomology, in the sense of [1], of algebraic hulls of G. This is accomplished by using the known structure theory for complex analytic linear groups in combination with certain easily established results concerning the cohomology of semidirect products. The main results are Theorem 4.1 (whose hypothesis is always satisfied by a complex analytic linear group) and Theorems 5.1 and 5.2. These last two theorems show that the usual abundantly used connections between complex analytic representations of complex analytic groups and rational representations of algebraic groups extend fully to the superstructure of cohomology.

Two-Dimensional Linear Groups Over Local Rings

1969 ◽  
Vol 21 ◽  
pp. 106-135 ◽  
Author(s):  
Norbert H. J. Lacroix
Keyword(s):  
Local Ring ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
Local Rings ◽  
Two Dimensional ◽  
Normal Subgroups ◽  
Field Case

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .

On the contractibility to a point of the linear groups of reflexive non-commutative Lp-spaces

1996 ◽  
Vol 119 (3) ◽  
pp. 545-560 ◽  
Author(s):  
Sergei V. Ferleger ◽  
Fyodor A. Sukochev
Keyword(s):  
Banach Space ◽  
Linear Group ◽  
Operator Norm ◽  
Linear Groups ◽  
Lp Spaces ◽  
Continuous Map

For every Banach space X, denote by GL(X) the linear group of X, i.e. the group of all linear continuous invertible operators on X with the topology induced by the operator norm. One says that GL(X) is contractible to a point if there exists a continuous map F: GL(X) × [0, 1] → GL(X) such that F(A,0) = A and F(A, 1) = Id, for every A ∈ GL(X).

scholarly journals Anti-integral extensions $ {R[{\alpha}]/R$

2004 ◽  
Vol 35 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Kiyoshi Baba ◽  
Ken-Ichi Yoshida
Keyword(s):  
Positive Integer ◽  
Integral Domain ◽  
Arbitrary Positive Integer ◽  
Integral Element ◽  
The Ideal

Let $ R $ be an integral domain and $ \alpha $ an anti-integral element of degree $ d $ over $ R $. In the paper [3] we give a condition for $ \alpha^2-a$ to be a unit of $ R[\alpha] $. In this paper we will generalize the result to an arbitrary positive integer $n$ and give a condition, in terms of the ideal $ I_{[\alpha]}^{n}D(\sqrt[n]{a}) $ of $ R $, for $ \alpha^{n}-a$ to be a unit of $ R[\alpha] $.

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