{"title":"Beauville p-Groups: A Survey","authors":"B. Fairbairn","doi":"10.1017/9781108692397.012","DOIUrl":"https://doi.org/10.1017/9781108692397.012","url":null,"abstract":"Beauville surfaces are a class of complex surfaces defined by letting a finite group \u0000G act on a product of Riemann surfaces. These surfaces possess many attractive \u0000geometric properties several of which are dictated by properties of the group G. \u0000In this survey we discuss the p-groups that may be used in this way. En route we \u0000discuss several open problems, questions and conjectures.","PeriodicalId":148530,"journal":{"name":"Groups St Andrews 2017 in Birmingham","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126104053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Subgroups of Twisted Wreath Products","authors":"P. Pálfy","doi":"10.1017/9781108692397.020","DOIUrl":"https://doi.org/10.1017/9781108692397.020","url":null,"abstract":"By determining subdirect products invariant under the action of a regular permutation group of the components we provide a natural motivation for the definition of twisted wreath products. Then—based on papers of R. Baddeley, A. Lucchini, F. B¨orner, and M. Aschbacher—we explain how twisted wreath products play a fundamental role in the problem of representing finite lattices as intervals in subgroup lattices of finite groups.","PeriodicalId":148530,"journal":{"name":"Groups St Andrews 2017 in Birmingham","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129049637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Aspherical Relative Presentations all Over Again","authors":"W. A. Bogley, M. Edjvet, Gerald Williams","doi":"10.1017/9781108692397.008","DOIUrl":"https://doi.org/10.1017/9781108692397.008","url":null,"abstract":"The concept of asphericity for relative group presentations was introduced twenty five years ago. Since then, the subject has advanced and detailed asphericity classifications have been obtained for various families of one-relator relative presentations. Through this work the definition of asphericity has evolved and new applications have emerged. In this article we bring together key results on relative asphericity, update them, and exhibit them under a single set of definitions and terminology. We describe consequences of asphericity and present techniques for proving asphericity and for proving non-asphericity. We give a detailed survey of results concerning one-relator relative presentations where the relator has free product length four.","PeriodicalId":148530,"journal":{"name":"Groups St Andrews 2017 in Birmingham","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123490527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Practical Computation with Linear Groups Over Infinite Domains","authors":"A. Detinko, D. Flannery","doi":"10.1017/9781108692397.011","DOIUrl":"https://doi.org/10.1017/9781108692397.011","url":null,"abstract":"We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the algorithms have been used to perform extensive computer experiments.","PeriodicalId":148530,"journal":{"name":"Groups St Andrews 2017 in Birmingham","volume":"205 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134165146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local-Global Conjectures and Blocks of Simple Groups","authors":"G. Malle, R. Kessar","doi":"10.1017/9781108692397.004","DOIUrl":"https://doi.org/10.1017/9781108692397.004","url":null,"abstract":"We give an expanded treatment of our lecture series at the 2017 Groups St Andrews conference in Birmingham on local-global conjectures and the block theory of finite reductive groups.","PeriodicalId":148530,"journal":{"name":"Groups St Andrews 2017 in Birmingham","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124619809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective","authors":"H. Helfgott","doi":"10.1017/9781108692397.014","DOIUrl":"https://doi.org/10.1017/9781108692397.014","url":null,"abstract":"By now, we have a product theorem in every finite simple group $G$ of Lie type, with the strength of the bound depending only in the rank of $G$. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Alt_n, we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem. \u0000We shall revisit the proof of the bound for Alt_n, bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a product theorem for Alt_n -- not of full strength, as that would be impossible, but strong enough to imply the diameter bound.","PeriodicalId":148530,"journal":{"name":"Groups St Andrews 2017 in Birmingham","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131431652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Irreducible Subgroups of Simple Algebraic Groups – A Survey","authors":"Timothy C. Burness, D. Testerman","doi":"10.1017/9781108692397.010","DOIUrl":"https://doi.org/10.1017/9781108692397.010","url":null,"abstract":"Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module. We say that $(G,H,V)$ is an irreducible triple if $V$ is irreducible as a $KH$-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups in characteristic zero to all algebraically closed fields. In this article we will survey recent advances towards a classification of irreducible triples for all positive dimensional subgroups of simple algebraic groups.","PeriodicalId":148530,"journal":{"name":"Groups St Andrews 2017 in Birmingham","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130531196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Simple Groups, Generation and Probabilistic Methods","authors":"Timothy C. Burness","doi":"10.1017/9781108692397.009","DOIUrl":"https://doi.org/10.1017/9781108692397.009","url":null,"abstract":"It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the extraordinary generation properties of simple groups, focussing on topics such as random generation, $(a,b)$-generation and spread, as well as highlighting the application of probabilistic methods in the proofs of many of the main results. We also present some recent work on the minimal generation of maximal and second maximal subgroups of simple groups, which has applications to the study of subgroup growth and the generation of primitive permutation groups.","PeriodicalId":148530,"journal":{"name":"Groups St Andrews 2017 in Birmingham","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128563643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Finite and Infinite Quotients of Discrete and Indiscrete Groups","authors":"P. Caprace","doi":"10.1017/9781108692397.003","DOIUrl":"https://doi.org/10.1017/9781108692397.003","url":null,"abstract":"These notes are devoted to lattices in products of trees and related topics. They provide an introduction to the construction, by M. Burger and S. Mozes, of examples of such lattices that are simple as abstract groups. Two features of that construction are emphasized: the relevance of non-discrete locally compact groups, and the two-step strategy in the proof of simplicity, addressing separately, and with completely different methods, the existence of finite and infinite quotients. A brief history of the quest for finitely generated and finitely presented infinite simple groups is also sketched. A comparison with Margulis' proof of Kneser's simplicity conjecture is discussed, and the relevance of the Classification of the Finite Simple Groups is pointed out. A final chapter is devoted to finite and infinite quotients of hyperbolic groups and their relation to the asymptotic properties of the finite simple groups. Numerous open problems are discussed along the way.","PeriodicalId":148530,"journal":{"name":"Groups St Andrews 2017 in Birmingham","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122644248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Character Degrees of a Sylow p-Subgroup of a Finite Chevalley Group G(pf) Over a Bad Prime","authors":"Tung Le, K. Magaard, A. Paolini","doi":"10.1017/9781108692397.018","DOIUrl":"https://doi.org/10.1017/9781108692397.018","url":null,"abstract":"Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G(q)$ defined over the field with $q$ elements. We first give a parametrization of the set $text{Irr}(U(q))$ of irreducible characters of $U(q)$ when $G(q)$ is of type $mathrm{G}_2$. This is uniform for primes $p ge 5$, while the bad primes $p=2$ and $p=3$ have to be considered separately. We then use this result and the contribution of several authors to show a general result, namely that if $G(q)$ is any finite Chevalley group with $p$ a bad prime, then there exists a character $chi in text{Irr}(U(q))$ such that $chi(1)=q^n/p$ for some $n in mathbb{Z}_{ge_0}$. In particular, for each $G(q)$ and every bad prime $p$, we construct a family of characters of such degree as inflation followed by an induction of linear characters of an abelian subquotient $V(q)$ of $U(q)$.","PeriodicalId":148530,"journal":{"name":"Groups St Andrews 2017 in Birmingham","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127952441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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