{"date_updated":"2024-05-07T08:30:29Z","language":[{"iso":"eng"}],"publication":"arXiv:2209.07927","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 interprete 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"}],"title":"Transitivity in finite general linear groups","author":[{"id":"46953","first_name":"Alena","full_name":"Ernst, Alena","last_name":"Ernst"},{"last_name":"Schmidt","full_name":"Schmidt, Kai-Uwe","first_name":"Kai-Uwe"}],"year":"2022","user_id":"46953","_id":"53534","status":"public","date_created":"2024-04-17T12:26:51Z","external_id":{"arxiv":["2209.07927"]},"citation":{"apa":"Ernst, A., &#38; Schmidt, K.-U. (2022). Transitivity in finite general linear groups. In <i>arXiv:2209.07927</i>.","mla":"Ernst, Alena, and Kai-Uwe Schmidt. “Transitivity in Finite General Linear Groups.” <i>ArXiv:2209.07927</i>, 2022.","chicago":"Ernst, Alena, and Kai-Uwe Schmidt. “Transitivity in Finite General Linear Groups.” <i>ArXiv:2209.07927</i>, 2022.","ama":"Ernst A, Schmidt K-U. Transitivity in finite general linear groups. <i>arXiv:220907927</i>. Published online 2022.","bibtex":"@article{Ernst_Schmidt_2022, title={Transitivity in finite general linear groups}, journal={arXiv:2209.07927}, author={Ernst, Alena and Schmidt, Kai-Uwe}, year={2022} }","ieee":"A. Ernst and K.-U. Schmidt, “Transitivity in finite general linear groups,” <i>arXiv:2209.07927</i>. 2022.","short":"A. Ernst, K.-U. Schmidt, ArXiv:2209.07927 (2022)."},"type":"preprint"}