[{"issue":"45","year":"2024","intvolume":"       307","citation":{"apa":"Ernst, A., &#38; Schmidt, K.-U. (2024). Transitivity in finite general linear groups. <i>Mathematische Zeitschrift</i>, <i>307</i>(45). <a href=\"https://doi.org/10.1007/s00209-024-03511-x\">https://doi.org/10.1007/s00209-024-03511-x</a>","short":"A. Ernst, K.-U. Schmidt, Mathematische Zeitschrift 307 (2024).","mla":"Ernst, Alena, and Kai-Uwe Schmidt. “Transitivity in Finite General Linear Groups.” <i>Mathematische Zeitschrift</i>, vol. 307, no. 45, 2024, doi:<a href=\"https://doi.org/10.1007/s00209-024-03511-x\">10.1007/s00209-024-03511-x</a>.","bibtex":"@article{Ernst_Schmidt_2024, title={Transitivity in finite general linear groups}, volume={307}, DOI={<a href=\"https://doi.org/10.1007/s00209-024-03511-x\">10.1007/s00209-024-03511-x</a>}, number={45}, journal={Mathematische Zeitschrift}, author={Ernst, Alena and Schmidt, Kai-Uwe}, year={2024} }","ama":"Ernst A, Schmidt K-U. Transitivity in finite general linear groups. <i>Mathematische Zeitschrift</i>. 2024;307(45). doi:<a href=\"https://doi.org/10.1007/s00209-024-03511-x\">10.1007/s00209-024-03511-x</a>","chicago":"Ernst, Alena, and Kai-Uwe Schmidt. “Transitivity in Finite General Linear Groups.” <i>Mathematische Zeitschrift</i> 307, no. 45 (2024). <a href=\"https://doi.org/10.1007/s00209-024-03511-x\">https://doi.org/10.1007/s00209-024-03511-x</a>.","ieee":"A. Ernst and K.-U. Schmidt, “Transitivity in finite general linear groups,” <i>Mathematische Zeitschrift</i>, vol. 307, no. 45, 2024, doi: <a href=\"https://doi.org/10.1007/s00209-024-03511-x\">10.1007/s00209-024-03511-x</a>."},"date_updated":"2024-06-17T10:04:29Z","volume":307,"author":[{"first_name":"Alena","id":"46953","full_name":"Ernst, Alena","last_name":"Ernst"},{"full_name":"Schmidt, Kai-Uwe","last_name":"Schmidt","first_name":"Kai-Uwe"}],"date_created":"2024-04-17T12:26:51Z","title":"Transitivity in finite general linear groups","doi":"10.1007/s00209-024-03511-x","publication":"Mathematische Zeitschrift","type":"journal_article","abstract":[{"text":"It is known that the notion of a transitive subgroup of a permutation group\r\n$G$ extends naturally to subsets of $G$. We consider subsets of the general\r\nlinear group $\\operatorname{GL}(n,q)$ acting transitively on flag-like\r\nstructures, which are common generalisations of $t$-dimensional subspaces of\r\n$\\mathbb{F}_q^n$ and bases of $t$-dimensional subspaces of $\\mathbb{F}_q^n$. We\r\ngive structural characterisations of transitive subsets of\r\n$\\operatorname{GL}(n,q)$ using the character theory of $\\operatorname{GL}(n,q)$\r\nand interpret such subsets as designs in the conjugacy class association\r\nscheme of $\\operatorname{GL}(n,q)$. In particular we generalise a theorem of\r\nPerin on subgroups of $\\operatorname{GL}(n,q)$ acting transitively on\r\n$t$-dimensional subspaces. We survey transitive subgroups of\r\n$\\operatorname{GL}(n,q)$, showing that there is no subgroup of\r\n$\\operatorname{GL}(n,q)$ with $1<t<n$ acting transitively on $t$-dimensional\r\nsubspaces unless it contains $\\operatorname{SL}(n,q)$ or is one of two\r\nexceptional groups. On the other hand, for all fixed $t$, we show that there\r\nexist nontrivial subsets of $\\operatorname{GL}(n,q)$ that are transitive on\r\nlinearly independent $t$-tuples of $\\mathbb{F}_q^n$, which also shows the\r\nexistence of nontrivial subsets of $\\operatorname{GL}(n,q)$ that are transitive\r\non more general flag-like structures. We establish connections with orthogonal\r\npolynomials, namely the Al-Salam-Carlitz polynomials, and generalise a result\r\nby Rudvalis and Shinoda on the distribution of the number of fixed points of\r\nthe elements in $\\operatorname{GL}(n,q)$. Many of our results can be\r\ninterpreted as $q$-analogs of corresponding results for the symmetric group.","lang":"eng"}],"status":"public","_id":"53534","department":[{"_id":"100"}],"user_id":"46953","language":[{"iso":"eng"}]},{"volume":175,"date_created":"2024-04-17T12:23:18Z","author":[{"first_name":"Alena","last_name":"Ernst","full_name":"Ernst, Alena","id":"46953"},{"first_name":"Kai-Uwe","last_name":"Schmidt","full_name":"Schmidt, Kai-Uwe"}],"publisher":"Cambridge University Press (CUP)","date_updated":"2024-05-07T08:29:59Z","doi":"10.1017/s0305004123000075","title":"Intersection theorems for finite general linear groups","issue":"1","publication_identifier":{"issn":["0305-0041","1469-8064"]},"publication_status":"published","page":"129-160","intvolume":"       175","citation":{"apa":"Ernst, A., &#38; Schmidt, K.-U. (2023). Intersection theorems for finite general linear groups. <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, <i>175</i>(1), 129–160. <a href=\"https://doi.org/10.1017/s0305004123000075\">https://doi.org/10.1017/s0305004123000075</a>","mla":"Ernst, Alena, and Kai-Uwe Schmidt. “Intersection Theorems for Finite General Linear Groups.” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, vol. 175, no. 1, Cambridge University Press (CUP), 2023, pp. 129–60, doi:<a href=\"https://doi.org/10.1017/s0305004123000075\">10.1017/s0305004123000075</a>.","bibtex":"@article{Ernst_Schmidt_2023, title={Intersection theorems for finite general linear groups}, volume={175}, DOI={<a href=\"https://doi.org/10.1017/s0305004123000075\">10.1017/s0305004123000075</a>}, number={1}, journal={Mathematical Proceedings of the Cambridge Philosophical Society}, publisher={Cambridge University Press (CUP)}, author={Ernst, Alena and Schmidt, Kai-Uwe}, year={2023}, pages={129–160} }","short":"A. Ernst, K.-U. Schmidt, Mathematical Proceedings of the Cambridge Philosophical Society 175 (2023) 129–160.","ama":"Ernst A, Schmidt K-U. Intersection theorems for finite general linear groups. <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. 2023;175(1):129-160. doi:<a href=\"https://doi.org/10.1017/s0305004123000075\">10.1017/s0305004123000075</a>","ieee":"A. Ernst and K.-U. Schmidt, “Intersection theorems for finite general linear groups,” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, vol. 175, no. 1, pp. 129–160, 2023, doi: <a href=\"https://doi.org/10.1017/s0305004123000075\">10.1017/s0305004123000075</a>.","chicago":"Ernst, Alena, and Kai-Uwe Schmidt. “Intersection Theorems for Finite General Linear Groups.” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i> 175, no. 1 (2023): 129–60. <a href=\"https://doi.org/10.1017/s0305004123000075\">https://doi.org/10.1017/s0305004123000075</a>."},"year":"2023","department":[{"_id":"100"}],"user_id":"46953","_id":"53533","language":[{"iso":"eng"}],"keyword":["General Mathematics"],"publication":"Mathematical Proceedings of the Cambridge Philosophical Society","type":"journal_article","status":"public"}]