---
_id: '53534'
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 $1<t<n$ 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."
author:
- first_name: Alena
  full_name: Ernst, Alena
  id: '46953'
  last_name: Ernst
- first_name: Kai-Uwe
  full_name: Schmidt, Kai-Uwe
  last_name: Schmidt
citation:
  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>
  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>
  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} }'
  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>.'
  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>.
  short: A. Ernst, K.-U. Schmidt, Mathematische Zeitschrift 307 (2024).
date_created: 2024-04-17T12:26:51Z
date_updated: 2024-06-17T10:04:29Z
department:
- _id: '100'
doi: 10.1007/s00209-024-03511-x
intvolume: '       307'
issue: '45'
language:
- iso: eng
publication: Mathematische Zeitschrift
status: public
title: Transitivity in finite general linear groups
type: journal_article
user_id: '46953'
volume: 307
year: '2024'
...
---
_id: '53533'
author:
- first_name: Alena
  full_name: Ernst, Alena
  id: '46953'
  last_name: Ernst
- first_name: Kai-Uwe
  full_name: Schmidt, Kai-Uwe
  last_name: Schmidt
citation:
  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>
  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>
  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} }'
  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>.'
  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>.'
  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>.
  short: A. Ernst, K.-U. Schmidt, Mathematical Proceedings of the Cambridge Philosophical
    Society 175 (2023) 129–160.
date_created: 2024-04-17T12:23:18Z
date_updated: 2024-05-07T08:29:59Z
department:
- _id: '100'
doi: 10.1017/s0305004123000075
intvolume: '       175'
issue: '1'
keyword:
- General Mathematics
language:
- iso: eng
page: 129-160
publication: Mathematical Proceedings of the Cambridge Philosophical Society
publication_identifier:
  issn:
  - 0305-0041
  - 1469-8064
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Intersection theorems for finite general linear groups
type: journal_article
user_id: '46953'
volume: 175
year: '2023'
...