{"citation":{"ieee":"A. Ernst and K.-U. Schmidt, “Transitivity in finite general linear groups,” Mathematische Zeitschrift, vol. 307, no. 45, 2024, doi: 10.1007/s00209-024-03511-x.","bibtex":"@article{Ernst_Schmidt_2024, title={Transitivity in finite general linear groups}, volume={307}, DOI={10.1007/s00209-024-03511-x}, number={45}, journal={Mathematische Zeitschrift}, author={Ernst, Alena and Schmidt, Kai-Uwe}, year={2024} }","apa":"Ernst, A., & Schmidt, K.-U. (2024). Transitivity in finite general linear groups. Mathematische Zeitschrift, 307(45). https://doi.org/10.1007/s00209-024-03511-x","mla":"Ernst, Alena, and Kai-Uwe Schmidt. “Transitivity in Finite General Linear Groups.” Mathematische Zeitschrift, vol. 307, no. 45, 2024, doi:10.1007/s00209-024-03511-x.","short":"A. Ernst, K.-U. Schmidt, Mathematische Zeitschrift 307 (2024).","ama":"Ernst A, Schmidt K-U. Transitivity in finite general linear groups. Mathematische Zeitschrift. 2024;307(45). doi:10.1007/s00209-024-03511-x","chicago":"Ernst, Alena, and Kai-Uwe Schmidt. “Transitivity in Finite General Linear Groups.” Mathematische Zeitschrift 307, no. 45 (2024). https://doi.org/10.1007/s00209-024-03511-x."},"issue":"45","date_updated":"2024-06-17T10:04:29Z","date_created":"2024-04-17T12:26:51Z","status":"public","doi":"10.1007/s00209-024-03511-x","type":"journal_article","abstract":[{"lang":"eng","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