A Monte Carlo algorithm is presented for constructing the natural representation of a group G that is known to be isomorphic to GL(n, 2). The complexity parameters are the natural dimension n and the storage space required to represent an element of G. What is surprising about this result is that both the data structure used to compute the isomorphism and each invocation of the isomorphism require polynomial time complexity. The ultimate goal is to eventually extend this result to the larger question of constructing the natural representation of classical groups. Extensions of the methods developed in this paper are discussed as well as open questions.
{"title":"Constructive recognition of a black box group isomorphic to GL(n,2)","authors":"G. Cooperman, L. Finkelstein, S. Linton","doi":"10.1090/dimacs/028/07","DOIUrl":"https://doi.org/10.1090/dimacs/028/07","url":null,"abstract":"A Monte Carlo algorithm is presented for constructing the natural representation of a group G that is known to be isomorphic to GL(n, 2). The complexity parameters are the natural dimension n and the storage space required to represent an element of G. What is surprising about this result is that both the data structure used to compute the isomorphism and each invocation of the isomorphism require polynomial time complexity. The ultimate goal is to eventually extend this result to the larger question of constructing the natural representation of classical groups. Extensions of the methods developed in this paper are discussed as well as open questions.","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131383054","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":"Asymptotic results for simple groups and some applications","authors":"L. Pyber","doi":"10.1090/dimacs/028/18","DOIUrl":"https://doi.org/10.1090/dimacs/028/18","url":null,"abstract":"","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"2000 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116679882","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}
The goal of this work is to overcome the learning barriers faced when first using parallelism. Currently, in order to parallelize a system such as GAP, one must embed a message passing library such as MPI, with many routines and many parameters. GAP/MPI provides a simple, task-oriented interface sitting above the MPI library. The system presents the end-user with a single SPMD (single program, multiple data) environment in GAP: an existing, familiar interactive language. In GAP/MPI one describes the end application in terms of high level tasks, which are invoked by a single procedure call in GAP/MPI. This eliminates the complexities of a message passing library, such as encoding a message in a suitable data structure, message synchronization, communication topologies and deadlock avoidance.
{"title":"GAP/MPI: Facilitating parallelism","authors":"G. Cooperman","doi":"10.1090/dimacs/028/06","DOIUrl":"https://doi.org/10.1090/dimacs/028/06","url":null,"abstract":"The goal of this work is to overcome the learning barriers faced when first using parallelism. Currently, in order to parallelize a system such as GAP, one must embed a message passing library such as MPI, with many routines and many parameters. GAP/MPI provides a simple, task-oriented interface sitting above the MPI library. The system presents the end-user with a single SPMD (single program, multiple data) environment in GAP: an existing, familiar interactive language. In GAP/MPI one describes the end application in terms of high level tasks, which are invoked by a single procedure call in GAP/MPI. This eliminates the complexities of a message passing library, such as encoding a message in a suitable data structure, message synchronization, communication topologies and deadlock avoidance.","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115016309","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":"Asymptotic Results for Permutation Groups","authors":"L. Pyber","doi":"10.1090/dimacs/011/14","DOIUrl":"https://doi.org/10.1090/dimacs/011/14","url":null,"abstract":"","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126298797","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":"Calculating the order of an invertible matrix","authors":"F. Celler, C. Leedham-Green","doi":"10.1090/dimacs/028/04","DOIUrl":"https://doi.org/10.1090/dimacs/028/04","url":null,"abstract":"","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132787270","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":"Application of Group Theory to Combinatorial Searches","authors":"C. Lam","doi":"10.1090/dimacs/011/10","DOIUrl":"https://doi.org/10.1090/dimacs/011/10","url":null,"abstract":"","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"130 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134459381","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":"Cayley Graphs and Direct-Product Graphs","authors":"A. Rosenberg","doi":"10.1090/dimacs/011/16","DOIUrl":"https://doi.org/10.1090/dimacs/011/16","url":null,"abstract":"","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130100431","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}
A variety of elementary combinatorial techniques for permutation groups are reviewed. It is shown how to apply these techniques to yield faster and/or more space-efficient algorithms for problems including group membership, normal closure, center, base change and Cayley graphs. Emphasis is placed on randomized techniques and new data structures. The paper includes both a survey of recent algorithms with which the authors have been associated, and some new algorithms in the same spirit that have not previously appeared in print. Many of the results include both complexity bounds and pseudo-code, along with comments for faster software implementations.
{"title":"Combinatorial Tools for Computational Group Theory","authors":"G. Cooperman, L. Finkelstein","doi":"10.1090/dimacs/011/05","DOIUrl":"https://doi.org/10.1090/dimacs/011/05","url":null,"abstract":"A variety of elementary combinatorial techniques for permutation groups are reviewed. It is shown how to apply these techniques to yield faster and/or more space-efficient algorithms for problems including group membership, normal closure, center, base change and Cayley graphs. Emphasis is placed on randomized techniques and new data structures. The paper includes both a survey of recent algorithms with which the authors have been associated, and some new algorithms in the same spirit that have not previously appeared in print. Many of the results include both complexity bounds and pseudo-code, along with comments for faster software implementations.","PeriodicalId":342609,"journal":{"name":"Groups And Computation","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128783160","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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