Pub Date : 2023-10-02DOI: 10.1017/s0305004123000506
Adrien Le Boudec, Nicolás Matte Bon
Abstract We construct finitely generated torsion-free solvable groups G that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of G are virtually abelian. In particular all finitely generated metabelian subgroups of G are virtually abelian. The existence of such groups shows that there is no “torsion-free version” of P. Kropholler’s theorem, which characterises solvable groups of infinite rank via their metabelian subquotients.
{"title":"Some torsion-free solvable groups with few subquotients","authors":"Adrien Le Boudec, Nicolás Matte Bon","doi":"10.1017/s0305004123000506","DOIUrl":"https://doi.org/10.1017/s0305004123000506","url":null,"abstract":"Abstract We construct finitely generated torsion-free solvable groups G that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of G are virtually abelian. In particular all finitely generated metabelian subgroups of G are virtually abelian. The existence of such groups shows that there is no “torsion-free version” of P. Kropholler’s theorem, which characterises solvable groups of infinite rank via their metabelian subquotients.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135789980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-02DOI: 10.1017/s030500412300049x
ETHAN ACKELSBERG, VITALY BERGELSON
Abstract We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If K is a number field with ring of integers $mathcal{O}_K$ and $E subseteq mathcal{O}_K$ has positive upper Banach density $d^*(E) = delta > 0$ , we show, inter alia : (1) if $p(x) in K[x]$ is an intersective polynomial (i.e., p has a root modulo m for every $m in mathcal{O}_K$ ) with $p(mathcal{O}_K) subseteq mathcal{O}_K$ and $r, s in mathcal{O}_K$ are distinct and nonzero, then for any $varepsilon > 0$ , there is a syndetic set $S subseteq mathcal{O}_K$ such that for any $n in S$ , begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + rp(n), x + sp(n)} subseteq E right} right) > delta^3 - varepsilon. end{align*} Moreover, if ${s}/{r} in mathbb{Q}$ , then there are syndetically many $n in mathcal{O}_K$ such that begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + rp(n), x + sp(n), x + (r+s)p(n)} subseteq E right} right) > delta^4 - varepsilon; end{align*} (2) if ${p_1, dots, p_k} subseteq K[x]$ is a jointly intersective family (i.e., $p_1, dots, p_k$ have a common root modulo m for every $m in mathcal{O}_K$ ) of linearly independent polynomials with $p_i(mathcal{O}_K) subseteq mathcal{O}_K$ , then there are syndetically many $n in mathcal{O}_K$ such that begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + p_1(n), dots, x + p_k(n)} subseteq E right} right) > delta^{k+1} - varepsilon. end{align*} These two results generalise and extend previous work of Frantzikinakis and Kra [ 21 ] and Franztikinakis [ 19 ] on polynomial configurations in $mathbb{Z}$ and build upon recent work of the authors and Best [ 2 ] on linear patterns in general abelian groups. The above combinatorial results follow from multiple recurrence results in ergodic theory via a version of Furstenberg’s correspondence principle. The ergodic-theoretic recurrence theorems require a sharpening of existing tools for handling polynomial multiple ergodic averages. A key advancement made in this paper is a new result on the equidistribution of polynomial orbits in nilmanifolds, which can be seen as a far-reaching generalisation of Weyl’s equidistribution theorem for polynomials of several variables: (3) let $d, k, l in mathbb{N}$ . Let $(X, mathcal{B}, mu, T_1, dots, T_l)$ be an ergodic, connected $mathbb{Z}^l$ -nilsystem. Let ${p_{i,j} ;:; 1 le i le k, 1 le j le l} subseteq mathbb{Q}[x_1, dots, x_d]$ be a family of polynomials such that $p_{i,j}left( mathbb{Z}^d right) subseteq mathbb{Z}$ and ${1} cup {p_{i,j}}$ is linearly independent over $mathbb{Q}$ . Then the $mathbb{Z}^d$ -sequence $left( prod_{j=1}^l{T_j^{p_{1,j}(n)}}x, dots, prod_{j=1}^l{T_j^{p_{k,j}(n)}}x right)_{n in mathbb{Z}^d}$ is well-distributed in $X^k$ for every x in a co-meager set of full measure.
摘要证明了数域整数环上多项式模式自然族的普遍差异现象(即大交集现象)。如果K是一个带整数环的数字域 $mathcal{O}_K$ 和 $E subseteq mathcal{O}_K$ 上巴拿赫密度是正的 $d^*(E) = delta > 0$ ，我们表明，除其他外:(1)如果 $p(x) in K[x]$ 是否一个相交多项式(即p对每一个 $m in mathcal{O}_K$ )与 $p(mathcal{O}_K) subseteq mathcal{O}_K$ 和 $r, s in mathcal{O}_K$ 都是不同且非零的 $varepsilon > 0$ ，有一个综合集 $S subseteq mathcal{O}_K$ 这样对于任何 $n in S$ ， begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + rp(n), x + sp(n)} subseteq E right} right) > delta^3 - varepsilon. end{align*} 此外，如果 ${s}/{r} in mathbb{Q}$ ，那么总共就有很多 $n in mathcal{O}_K$ 这样 begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + rp(n), x + sp(n), x + (r+s)p(n)} subseteq E right} right) > delta^4 - varepsilon; end{align*} (2)如果 ${p_1, dots, p_k} subseteq K[x]$ 是一个共同相交的家族(即 $p_1, dots, p_k$ 对m取模有公根吗 $m in mathcal{O}_K$ 的线性无关多项式 $p_i(mathcal{O}_K) subseteq mathcal{O}_K$ ，那么总共就有很多 $n in mathcal{O}_K$ 这样 begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + p_1(n), dots, x + p_k(n)} subseteq E right} right) > delta^{k+1} - varepsilon. end{align*} 这两个结果推广和扩展了Frantzikinakis和Kra[21]和Franztikinakis[19]关于多项式构型的研究 $mathbb{Z}$ 并以作者和Best[2]最近关于一般阿贝尔群的线性模式的工作为基础。上述组合结果是由遍历理论中的多次递归结果通过弗斯滕伯格对应原理的一个版本推导出来的。遍历理论递归定理要求对现有的处理多项式多次遍历平均的工具进行改进。本文的一个关键进展是关于零流形中多项式轨道的均衡分布的一个新结果，这可以看作是Weyl的多变量多项式均衡分布定理的一个深远推广 $d, k, l in mathbb{N}$ ． 让 $(X, mathcal{B}, mu, T_1, dots, T_l)$ 做一个通情达理的人 $mathbb{Z}^l$ -零系统。让 ${p_{i,j} ;:; 1 le i le k, 1 le j le l} subseteq mathbb{Q}[x_1, dots, x_d]$ 是一个多项式族，这样 $p_{i,j}left( mathbb{Z}^d right) subseteq mathbb{Z}$ 和 ${1} cup {p_{i,j}}$ 是线性无关的 $mathbb{Q}$ ． 然后是 $mathbb{Z}^d$ -序列 $left( prod_{j=1}^l{T_j^{p_{1,j}(n)}}x, dots, prod_{j=1}^l{T_j^{p_{k,j}(n)}}x right)_{n in mathbb{Z}^d}$ 分布在 $X^k$ 对于全测度集合中的每一个x。
{"title":"Multiple recurrence and popular differences for polynomial patterns in rings of integers","authors":"ETHAN ACKELSBERG, VITALY BERGELSON","doi":"10.1017/s030500412300049x","DOIUrl":"https://doi.org/10.1017/s030500412300049x","url":null,"abstract":"Abstract We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If K is a number field with ring of integers $mathcal{O}_K$ and $E subseteq mathcal{O}_K$ has positive upper Banach density $d^*(E) = delta > 0$ , we show, inter alia : (1) if $p(x) in K[x]$ is an intersective polynomial (i.e., p has a root modulo m for every $m in mathcal{O}_K$ ) with $p(mathcal{O}_K) subseteq mathcal{O}_K$ and $r, s in mathcal{O}_K$ are distinct and nonzero, then for any $varepsilon > 0$ , there is a syndetic set $S subseteq mathcal{O}_K$ such that for any $n in S$ , begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + rp(n), x + sp(n)} subseteq E right} right) > delta^3 - varepsilon. end{align*} Moreover, if ${s}/{r} in mathbb{Q}$ , then there are syndetically many $n in mathcal{O}_K$ such that begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + rp(n), x + sp(n), x + (r+s)p(n)} subseteq E right} right) > delta^4 - varepsilon; end{align*} (2) if ${p_1, dots, p_k} subseteq K[x]$ is a jointly intersective family (i.e., $p_1, dots, p_k$ have a common root modulo m for every $m in mathcal{O}_K$ ) of linearly independent polynomials with $p_i(mathcal{O}_K) subseteq mathcal{O}_K$ , then there are syndetically many $n in mathcal{O}_K$ such that begin{align*} d^* left( left{ x in mathcal{O}_K ;:; {x, x + p_1(n), dots, x + p_k(n)} subseteq E right} right) > delta^{k+1} - varepsilon. end{align*} These two results generalise and extend previous work of Frantzikinakis and Kra [ 21 ] and Franztikinakis [ 19 ] on polynomial configurations in $mathbb{Z}$ and build upon recent work of the authors and Best [ 2 ] on linear patterns in general abelian groups. The above combinatorial results follow from multiple recurrence results in ergodic theory via a version of Furstenberg’s correspondence principle. The ergodic-theoretic recurrence theorems require a sharpening of existing tools for handling polynomial multiple ergodic averages. A key advancement made in this paper is a new result on the equidistribution of polynomial orbits in nilmanifolds, which can be seen as a far-reaching generalisation of Weyl’s equidistribution theorem for polynomials of several variables: (3) let $d, k, l in mathbb{N}$ . Let $(X, mathcal{B}, mu, T_1, dots, T_l)$ be an ergodic, connected $mathbb{Z}^l$ -nilsystem. Let ${p_{i,j} ;:; 1 le i le k, 1 le j le l} subseteq mathbb{Q}[x_1, dots, x_d]$ be a family of polynomials such that $p_{i,j}left( mathbb{Z}^d right) subseteq mathbb{Z}$ and ${1} cup {p_{i,j}}$ is linearly independent over $mathbb{Q}$ . Then the $mathbb{Z}^d$ -sequence $left( prod_{j=1}^l{T_j^{p_{1,j}(n)}}x, dots, prod_{j=1}^l{T_j^{p_{k,j}(n)}}x right)_{n in mathbb{Z}^d}$ is well-distributed in $X^k$ for every x in a co-meager set of full measure.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"304 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135830931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-28DOI: 10.1017/s0305004123000476
Jonas Bergström, Everett W. Howe, Elisa Lorenzo García, Christophe Ritzenthaler
Abstract For a given genus $g geq 1$ , we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $mathbb{F}_q$ . As a consequence of Katz–Sarnak theory, we first get for any given $g>0$ , any $varepsilon>0$ and all q large enough, the existence of a curve of genus g over $mathbb{F}_q$ with at least $1+q+ (2g-varepsilon) sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 sqrt{q}$ valid for $g geq 3$ and odd $q geq 11$ . Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 sqrt{q} -32$ is valid for all $gge 2$ and for all q .
{"title":"Lower bounds on the maximal number of rational points on curves over finite fields","authors":"Jonas Bergström, Everett W. Howe, Elisa Lorenzo García, Christophe Ritzenthaler","doi":"10.1017/s0305004123000476","DOIUrl":"https://doi.org/10.1017/s0305004123000476","url":null,"abstract":"Abstract For a given genus $g geq 1$ , we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $mathbb{F}_q$ . As a consequence of Katz–Sarnak theory, we first get for any given $g>0$ , any $varepsilon>0$ and all q large enough, the existence of a curve of genus g over $mathbb{F}_q$ with at least $1+q+ (2g-varepsilon) sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 sqrt{q}$ valid for $g geq 3$ and odd $q geq 11$ . Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 sqrt{q} -32$ is valid for all $gge 2$ and for all q .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135343288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-28DOI: 10.1017/s0305004123000452
Philip Hackney
Abstract We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalised operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms. This allows for straightforward comparisons, and we use this to show that certain free-forgetful adjunctions between categories of generalised operads can be realised at the level of presheaves. This includes adjunctions between operads and cyclic operads, between dioperads and augmented cyclic operads, and between wheeled properads and modular operads.
{"title":"Categories of graphs for operadic structures","authors":"Philip Hackney","doi":"10.1017/s0305004123000452","DOIUrl":"https://doi.org/10.1017/s0305004123000452","url":null,"abstract":"Abstract We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalised operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms. This allows for straightforward comparisons, and we use this to show that certain free-forgetful adjunctions between categories of generalised operads can be realised at the level of presheaves. This includes adjunctions between operads and cyclic operads, between dioperads and augmented cyclic operads, and between wheeled properads and modular operads.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135342994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-27DOI: 10.1017/s0305004123000440
LINYI CHEN, GRANT CRIDER-PHILLIPS, BRAEDEN REINOSO, JOSHUA SABLOFF, LEYU YAO
Abstract We investigate when a Legendrian knot in the standard contact ${{mathbb{R}}}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we completely determine when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings and the minimisation of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting.
{"title":"Non-Orientable Lagrangian Fillings of Legendrian Knots","authors":"LINYI CHEN, GRANT CRIDER-PHILLIPS, BRAEDEN REINOSO, JOSHUA SABLOFF, LEYU YAO","doi":"10.1017/s0305004123000440","DOIUrl":"https://doi.org/10.1017/s0305004123000440","url":null,"abstract":"Abstract We investigate when a Legendrian knot in the standard contact ${{mathbb{R}}}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we completely determine when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings and the minimisation of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"326 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135539144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-06DOI: 10.1017/S0305004123000427
Sun-Kai Leung
Abstract Let $k geqslant 2$ be an integer. We prove that factorisation of integers into k parts follows the Dirichlet distribution $mathrm{Dir}left({1}/{k},ldots,{1}/{k}right)$ by multidimensional contour integration, thereby generalising the Deshouillers–Dress–Tenenbaum (DDT) arcsine law on divisors where $k=2$ . The same holds for factorisation of polynomials or permutations. Dirichlet distribution with arbitrary parameters can be modelled similarly.
{"title":"Dirichlet law for factorisation of integers, polynomials and permutations","authors":"Sun-Kai Leung","doi":"10.1017/S0305004123000427","DOIUrl":"https://doi.org/10.1017/S0305004123000427","url":null,"abstract":"Abstract Let \u0000$k geqslant 2$\u0000 be an integer. We prove that factorisation of integers into k parts follows the Dirichlet distribution \u0000$mathrm{Dir}left({1}/{k},ldots,{1}/{k}right)$\u0000 by multidimensional contour integration, thereby generalising the Deshouillers–Dress–Tenenbaum (DDT) arcsine law on divisors where \u0000$k=2$\u0000 . The same holds for factorisation of polynomials or permutations. Dirichlet distribution with arbitrary parameters can be modelled similarly.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"1 1","pages":"649 - 676"},"PeriodicalIF":0.8,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89251839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-04DOI: 10.1017/s0305004123000397
{"title":"PSP volume 175 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s0305004123000397","DOIUrl":"https://doi.org/10.1017/s0305004123000397","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"34 1","pages":"b1 - b2"},"PeriodicalIF":0.8,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89919205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-04DOI: 10.1017/s0305004123000385
{"title":"PSP volume 175 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s0305004123000385","DOIUrl":"https://doi.org/10.1017/s0305004123000385","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"569 1","pages":"f1 - f2"},"PeriodicalIF":0.8,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91373969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-10DOI: 10.1017/s030500412300035x
THOMAS MICHAEL KELLER, ALEXANDER MORETÓ
We prove that there exists a universal constant D such that if p is a prime divisor of the index of the Fitting subgroup of a finite group G, then the number of conjugacy classes of G is at least $Dp/log_2p$ . We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$ .
{"title":"Prime divisors and the number of conjugacy classes of finite groups","authors":"THOMAS MICHAEL KELLER, ALEXANDER MORETÓ","doi":"10.1017/s030500412300035x","DOIUrl":"https://doi.org/10.1017/s030500412300035x","url":null,"abstract":"We prove that there exists a universal constant <jats:italic>D</jats:italic> such that if <jats:italic>p</jats:italic> is a prime divisor of the index of the Fitting subgroup of a finite group <jats:italic>G</jats:italic>, then the number of conjugacy classes of <jats:italic>G</jats:italic> is at least <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030500412300035X_inline1.png\" /> <jats:tex-math> $Dp/log_2p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We conjecture that we can take <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030500412300035X_inline2.png\" /> <jats:tex-math> $D=1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and prove that for solvable groups, we can take <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030500412300035X_inline3.png\" /> <jats:tex-math> $D=1/3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"14 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-13DOI: 10.1017/s0305004123000312
{"title":"PSP volume 175 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s0305004123000312","DOIUrl":"https://doi.org/10.1017/s0305004123000312","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"263 1","pages":"b1 - b2"},"PeriodicalIF":0.8,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75771505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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