Some remarks on Lorenzen $r$-group of partly ordered groups

Jiří Močkoř; Angeliki Kontolatou

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Regularity of sets with constant intrinsic normal in a class of Carnot groups

Marco Marchi (2014)

Annales de l’institut Fourier

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In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type $☆$ ” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type $☆$ is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano...

Plane and layer $(p 2, 2^{l})$ symmetry groups

S. V. Jablan (1990)

Matematički Vesnik

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Plane and layer $(p^{p r i m e}, 2^{l})$ symmetry groups

S. V. Jablan (1990)

Matematički Vesnik

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A problem of Kollár and Larsen on finite linear groups and crepant resolutions

Robert Guralnick, Pham Tiep (2012)

Journal of the European Mathematical Society

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The notion of age of elements of complex linear groups was introduced by M. Reid and is of importance in algebraic geometry, in particular in the study of crepant resolutions and of quotients of Calabi–Yau varieties. In this paper, we solve a problem raised by J. Kollár and M. Larsen on the structure of finite irreducible linear groups generated by elements of age $\leq 1$ . More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation....

Groups of simple and multiple antisymmetry of similitude in $E^{2}$

S. V. Jablan (1986)

Matematički Vesnik

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Deformation theory and finite simple quotients of triangle groups I

Michael Larsen, Alexander Lubotzky, Claude Marion (2014)

Journal of the European Mathematical Society

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Let $2 \leq a \leq b \leq c \in ℕ$ with $μ = 1 / a + 1 / b + 1 / c < 1$ and let $T = T_{a, b, c} = 〈 x, y, z : x^{a} = y^{b} = z^{c} = x y z = 1 〉$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$ ? (Classically, for $(a, b, c) = (2, 3, 7)$ and more recently also for general $(a, b, c)$ .) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially...

A new characterization of Suzuki groups

Behnam Ebrahimzadeh, Reza Mohammadyari (2019)

Archivum Mathematicum

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One of the important questions that remains after the classification of the finite simple groups is how to recognize a simple group via specific properties. For example, authors have been able to use graphs associated to element orders and to number of elements with specific orders to determine simple groups up to isomorphism. In this paper, we prove that Suzuki groups $S z (q)$ , where $q \pm \sqrt{2 q} + 1$ is a prime number can be uniquely determined by the order of group and the number of elements with the same...

On the Davenport constant and group algebras

Daniel Smertnig (2010)

Colloquium Mathematicae

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For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S = g ₁ \cdot . . . \cdot g_{l}$ over G such that $(X^{g ₁} - a ₁) \cdot . . . \cdot (X^{g_{l}} - a_{l}) \neq 0 \in K [G]$ for all $a ₁, . . ., a_{l} \in K^{\times}$ . If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for...

$K (π, 1)$ conjecture for Artin groups

Luis Paris (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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The purpose of this paper is to put together a large amount of results on the $K (π, 1)$ conjecture for Artin groups, and to make them accessible to non-experts. Firstly, this is a survey, containing basic definitions, the main results, examples and an historical overview of the subject. But, it is also a reference text on the topic that contains proofs of a large part of the results on this question. Some proofs as well as few results are new. Furthermore, the text, being addressed to non-experts,...

On unit group of finite semisimple group algebras of non-metabelian groups up to order 72

Gaurav Mittal, Rajendra Kumar Sharma (2021)

Mathematica Bohemica

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We characterize the unit group of semisimple group algebras $𝔽_{q} G$ of some non-metabelian groups, where $F_{q}$ is a field with $q = p^{k}$ elements for $p$ prime and a positive integer $k$ . In particular, we consider all 6 non-metabelian groups of order 48, the only non-metabelian group $((C_{3} \times C_{3}) ⋊ C_{3}) ⋊ C_{2}$ of order 54, and 7 non-metabelian groups of order 72. This completes the study of unit groups of semisimple group algebras for groups upto order 72.

The density of representation degrees

Martin Liebeck, Dan Segal, Aner Shalev (2012)

Journal of the European Mathematical Society

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For a group $G$ and a positive real number $x$ , define $d_{G} (x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$ . We study the asymptotics of $d_{G} (x)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $α > 0$ such that $d_{G} (x) > x^{α}$ for all large $x$ , or $G$ is virtually abelian (in which case $d_{G} (x)$ is bounded). ...

$D_{p} (3)$ (p ≥ 5) can be characterized by its order components

Huaguo Shi, Zhangjia Han, Guiyun Chen (2012)

Colloquium Mathematicae

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Let G be a finite group, and $M = D_{p} (3)$ (p ≥ 3). It is proved that G ≅ M if G and M have the same order components.

Injectors of fitting classes of $\mathfrak{G}_{1}$ -groups

Federico Menegazzo, Martin L. Newell (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Fitting classes and injectors are discussed in the class of $\mathfrak{G}_{1}$ -groups. A necessary and sufficient condition for the existence of injectors is given; it is also shown that, when this condition holds, the injectors form a unique conjugacy class.

A duality theorem for Dieudonné displays

Eike Lau (2009)

Annales scientifiques de l'École Normale Supérieure

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We show that the Zink equivalence between $p$ -divisible groups and Dieudonné displays over a complete local ring with perfect residue field of characteristic $p$ is compatible with duality. The proof relies on a new explicit formula for the $p$ -divisible group associated to a Dieudonné display.

Presentations of finite simple groups: a computational approach

Robert Guralnick, William M. Kantor, Martin Kassabov, Alexander Lubotzky (2011)

Journal of the European Mathematical Society

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All finite simple groups of Lie type of rank $n$ over a field of size $q$ , with the possible exception of the Ree groups $^{2} G_{2} (q)$ , have presentations with at most 49 relations and bit-length $O (𝚕𝚘𝚐 n + 𝚕𝚘𝚐 q)$ . Moreover, $A_{n}$ and $S_{n}$ have presentations with 3 generators; 7 relations and bit-length $O (𝚕𝚘𝚐 n)$ , while $𝚂𝙻 (n, q)$ has a presentation with 6 generators, 25 relations and bit-length $O (𝚕𝚘𝚐 n + 𝚕𝚘𝚐 q)$ .

Coxeter group actions on the complement of hyperplanes and special involutions

Giovanni Felder, A. Veselov (2005)

Journal of the European Mathematical Society

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We consider both standard and twisted actions of a (real) Coxeter group $G$ on the complement $ℳ_{G}$ to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in $G$ and give explicit formulae which describe both actions on the total cohomology $H^{*} (ℳ_{G}, 𝒞)$ in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group $S_{n}$ , the...

Displaying similar documents to “Some remarks on Lorenzen r -group of partly ordered groups”

Displaying similar documents to “Some remarks on Lorenzen $r$ -group of partly ordered groups”