Presentations of finite simple groups: a computational approach

Robert Guralnick; William M. Kantor; Martin Kassabov; Alexander Lubotzky

Displaying similar documents to “Presentations of finite simple groups: a computational approach”

On the topology of polynomials with bounded integer coefficients

De-Jun Feng (2016)

Journal of the European Mathematical Society

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For a real number $q > 1$ and a positive integer $m$ , let $Y_{m} (q) : = \{\sum_{i = 0}^{n} ϵ_{i} q^{i} : ϵ_{i} \in \{0, \pm 1, ..., \pm m\}, n = 0, 1, ...\}$ . In this paper, we show that $Y_{m} (q)$ is dense in $ℝ$ if and only if $q < m + 1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].

Groups of given intermediate word growth

Laurent Bartholdi, Anna Erschler (2014)

Annales de l’institut Fourier

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We show that there exists a finitely generated group of growth $\sim f$ for all functions $f : ℝ_{+} \to ℝ_{+}$ satisfying $f (2 R) \leq f {(R)}^{2} \leq f (η_{+} R)$ for all $R$ large enough and $η_{+} \approx 2.4675$ the positive root of $X^{3} - X^{2} - 2 X - 4$ . Set $α_{-} = log 2 / log η_{+} \approx 0.7674$ ; then all functions that grow uniformly faster than $exp (R^{α_{-}})$ are realizable as the growth of a group. We also give a family of sum-contracting branched groups of growth $\sim exp (R^{α})$ for a dense set of $α \in [α_{-}, 1]$ .

$L_{p}$ inequalities for the growth of polynomials with restricted zeros

Nisar A. Rather, Suhail Gulzar, Aijaz A. Bhat (2022)

Archivum Mathematicum

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Let $P (z) = \sum_{ν = 0}^{n} a_{ν} z^{ν}$ be a polynomial of degree at most $n$ which does not vanish in the disk $| z | < 1$ , then for $1 \leq p < \infty$ and $R > 1$ , Boas and Rahman proved ${∥P (R z)∥}_{p} \leq ({∥R^{n} + z∥}_{p} / {∥1 + z∥}_{p}) {∥P∥}_{p} .$ In this paper, we improve the above inequality for $0 \leq p < \infty$ by involving some of the coefficients of the polynomial $P (z)$ . Analogous result for the class of polynomials $P (z)$ having no zero in $| z | > 1$ is also given.

The unit groups of semisimple group algebras of some non-metabelian groups of order $144$

Gaurav Mittal, Rajendra K. Sharma (2023)

Mathematica Bohemica

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We consider all the non-metabelian groups $G$ of order $144$ that have exponent either $36$ or $72$ and deduce the unit group $U (𝔽_{q} G)$ of semisimple group algebra $𝔽_{q} G$ . Here, $q$ denotes the power of a prime, i.e., $q = p^{r}$ for $p$ prime and a positive integer $r$ . Up to isomorphism, there are $6$ groups of order $144$ that have exponent either $36$ or $72$ . Additionally, we also discuss how to simply obtain the unit groups of the semisimple group algebras of those non-metabelian groups of order $144$ that are a direct product of two...

On the divisor function over Piatetski-Shapiro sequences

Hui Wang, Yu Zhang (2023)

Czechoslovak Mathematical Journal

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Let $[x]$ be an integer part of $x$ and $d (n)$ be the number of positive divisor of $n$ . Inspired by some results of M. Jutila (1987), we prove that for $1 < c < \frac{6}{5}$ , $\sum_{n \leq x} d ([n^{c}]) = c x log x + (2 γ - c) x + O (\frac{x}{log x}),$ where $γ$ is the Euler constant and $[n^{c}]$ is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.

Permutability of centre-by-finite groups

Brunetto Piochi (1989)

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

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Let $G$ be a group and $m$ be an integer greater than or equal to $2$ . $G$ is said to be $m$ -permutable if every product of $m$ elements can be reordered at least in one way. We prove that, if $G$ has a centre of finite index $z$ , then $G$ is $(1+[z/2])$ -permutable. More bounds are given on the least $m$ such that $G$ is $m$ -permutable.

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.

Global analytic and Gevrey surjectivity of the Mizohata operator $D_{2} + i x_{2}^{2 k} D_{1}$

Lamberto Cattabriga, Luisa Zanghirati (1990)

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

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The surjectivity of the operator $D_{2} + i x_{2}^{2 k} D_{1}$ from the Gevrey space $E^{\{s\}} (R^{2})$ , $s \geq 1$ , onto itself and its non-surjectivity from $E^{\{s\}} (R^{3})$ to $E^{\{s\}} (R^{3})$ is proved.

Limits of relatively hyperbolic groups and Lyndon’s completions

Olga Kharlampovich, Alexei Myasnikov (2012)

Journal of the European Mathematical Society

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We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon’s completion $G^{ℤ [t]}$ of the group $G$ , or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{ℤ [t]}$ containing $G$ is universally equivalent to $G$ . Since finitely...

Finite groups whose all proper subgroups are $𝒞$ -groups

Pengfei Guo, Jianjun Liu (2018)

Czechoslovak Mathematical Journal

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A group $G$ is said to be a $𝒞$ -group if for every divisor $d$ of the order of $G$ , there exists a subgroup $H$ of $G$ of order $d$ such that $H$ is normal or abnormal in $G$ . We give a complete classification of those groups which are not $𝒞$ -groups but all of whose proper subgroups are $𝒞$ -groups.