Counting arithmetic subgroups and subgroup growth of virtually free groups

Amichai Eisenmann

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Displaying similar documents to “Counting arithmetic subgroups and subgroup growth of virtually free groups”

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.

Mean values related to the Dedekind zeta-function

Hengcai Tang, Youjun Wang (2024)

Czechoslovak Mathematical Journal

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Let $K / ℚ$ be a nonnormal cubic extension which is given by an irreducible polynomial $g (x) = x^{3} + a x^{2} + b x + c$ . Denote by $ζ_{K} (s)$ the Dedekind zeta-function of the field $K$ and $a_{K} (n)$ the number of integral ideals in $K$ with norm $n$ . In this note, by the higher integral mean values and subconvexity bound of automorphic $L$ -functions, the second and third moment of $a_{K} (n)$ is considered, i.e., $\sum_{n \leq x} a_{K}^{2} (n) = x P_{1} (log x) + O (x^{5 / 7 + ϵ}), \sum_{n \leq x} a_{K}^{3} (n) = x P_{4} (log x) + O (X^{321 / 356 + ϵ}),$ where $P_{1} (t)$ , $P_{4} (t)$ are polynomials of degree 1, 4, respectively, $ϵ > 0$ is an arbitrarily small number.

Representation growth of linear groups

Michael Larsen, Alexander Lubotzky (2008)

Journal of the European Mathematical Society

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Let $Γ$ be a group and $r_{n} (Γ)$ the number of its $n$ -dimensional irreducible complex representations. We define and study the associated representation zeta function $𝒵_{Γ} (s) = \sum_{n = 1}^{\infty} r_{n} (Γ) n^{- s}$ . When $Γ$ is an arithmetic group satisfying the congruence subgroup property then $𝒵_{Γ} (s)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite...

On $σ$ -permutably embedded subgroups of finite groups

Chenchen Cao, Li Zhang, Wenbin Guo (2019)

Czechoslovak Mathematical Journal

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Let $σ = {σ_{i} : i \in I}$ be some partition of the set of all primes $ℙ$ , $G$ be a finite group and $σ (G) = {σ_{i} : σ_{i} \cap π (G) \neq \emptyset}$ . A set $ℋ$ of subgroups of $G$ is said to be a complete Hall $σ$ -set of $G$ if every non-identity member of $ℋ$ is a Hall $σ_{i}$ -subgroup of $G$ and $ℋ$ contains exactly one Hall $σ_{i}$ -subgroup of $G$ for every $σ_{i} \in σ (G)$ . $G$ is said to be $σ$ -full if $G$ possesses a complete Hall $σ$ -set. A subgroup $H$ of $G$ is $σ$ -permutable in $G$ if $G$ possesses a complete Hall $σ$ -set $ℋ$ such that $H A^{x}$ = $A^{x} H$ for all $A \in ℋ$ and all $x \in G$ . A subgroup $H$ of $G$ is $σ$ -permutably embedded in...

A note on infinite $a S$ -groups

Reza Nikandish, Babak Miraftab (2015)

Czechoslovak Mathematical Journal

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Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $a S$ -group. We study some properties of $a S$ -groups. For instance, it is shown that a nilpotent group $G$ is an $a S$ -group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $a S$ -group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

The Ribes-Zalesskii property of some one relator groups

Gilbert Mantika, Narcisse Temate-Tangang, Daniel Tieudjo (2022)

Archivum Mathematicum

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The profinite topology on any abstract group $G$ , is one such that the fundamental system of neighborhoods of the identity is given by all its subgroups of finite index. We say that a group $G$ has the Ribes-Zalesskii property of rank $k$ , or is RZ $_{k}$ with $k$ a natural number, if any product $H_{1} H_{2} \dots H_{k}$ of finitely generated subgroups $H_{1}, H_{2}, \dots, H_{k}$ is closed in the profinite topology on $G$ . And a group is said to have the Ribes-Zalesskii property or is RZ if it is RZ $_{k}$ for any natural number $k$ . In this paper we characterize...

On generalized square-full numbers in an arithmetic progression

Angkana Sripayap, Pattira Ruengsinsub, Teerapat Srichan (2022)

Czechoslovak Mathematical Journal

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Let $a$ and $b \in ℕ$ . Denote by $R_{a, b}$ the set of all integers $n > 1$ whose canonical prime representation $n = p_{1}^{α_{1}} p_{2}^{α_{2}} \dots p_{r}^{α_{r}}$ has all exponents $α_{i}$ $(1 \leq i \leq r)$ being a multiple of $a$ or belonging to the arithmetic progression $a t + b$ , $t \in ℕ_{0} : = ℕ \cup {0}$ . All integers in $R_{a, b}$ are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given. ...

Cambrian fans

Nathan Reading, David E. Speyer (2009)

Journal of the European Mathematical Society

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For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$ ; the $c$ -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$ . Its maximal cones are naturally indexed by the $c$ -sortable elements of $W$ . The main result of this paper is that the known bijection cl $_{c}$ between $c$ -sortable elements and $c$ -clusters induces a combinatorial isomorphism of fans. In particular, the $c$ -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for...

A new approach to construct uninorms via uninorms on bounded lattices

Zhen-Yu Xiu, Xu Zheng (2024)

Kybernetika

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In this paper, on a bounded lattice $L$ , we give a new approach to construct uninorms via a given uninorm $U^{*}$ on the subinterval $[0, a]$ (or $[b, 1]$ ) of $L$ under additional constraint conditions on $L$ and $U^{*}$ . This approach makes our methods generalize some known construction methods for uninorms in the literature. Meanwhile, some illustrative examples for the construction of uninorms on bounded lattices are provided.

Every $2$ -group with all subgroups normal-by-finite is locally finite

Enrico Jabara (2018)

Czechoslovak Mathematical Journal

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A group $G$ has all of its subgroups normal-by-finite if $H / H_{G}$ is finite for all subgroups $H$ of $G$ . The Tarski-groups provide examples of $p$ -groups ( $p$ a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a $2$ -group with every subgroup normal-by-finite is locally finite. We also prove that if $| H / H_{G} | \leq 2$ for every subgroup $H$ of $G$ , then $G$ contains an Abelian subgroup of index at most $8$ .

An explicit computation of $p$ -stabilized vectors

Michitaka MIYAUCHI, Takuya YAMAUCHI (2014)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we give a concrete method to compute $p$ -stabilized vectors in the space of parahori-fixed vectors for connected reductive groups over $p$ -adic fields. An application to the global setting is also discussed. In particular, we give an explicit $p$ -stabilized form of a Saito-Kurokawa lift.