{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T15:18:47Z","timestamp":1759936727621,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For a positive integer $k$, a $k$-relation on a set $\\Omega$ is a non-empty subset $\\Delta$ of the $k$-fold Cartesian product $\\Omega^k$; $\\Delta$ is called a $k$-relation for a permutation group $H$ on $\\Omega$ if $H$ leaves $\\Delta$ invariant setwise. The $k$-closure $H^{(k)}$ of  $H$, in the sense of Wielandt, is the largest permutation group $K$ on $\\Omega$ such that the set of $k$-relations for $K$ is equal to the set of $k$-relations for  $H$. We study $k$-relations for finite semi-linear groups $H\\leq{\\rm\\Gamma L}(d,q)$ in their natural action on the set $\\Omega$ of non-zero vectors of the underlying vector space. In particular, for each Aschbacher class ${\\mathcal C}$ of geometric subgroups of ${\\rm\\Gamma L}(d,q)$, we define a subset ${\\rm Rel}({\\mathcal C})$ of $k$-relations (with $k=1$ or $k=2$) and prove (i) that $H$ lies in ${\\mathcal C}$ if and only if $H$ leaves invariant at least one relation in ${\\rm Rel}({\\mathcal C})$, and (ii) that, if $H$ is maximal among subgroups in ${\\mathcal C}$, then an element $g\\in{\\rm\\Gamma L}(d,q)$ lies in the $k$-closure of $H$ if and only if $g$ leaves invariant a single $H$-invariant $k$-relation in ${\\rm Rel}({\\mathcal C})$ (rather than checking that $g$ leaves invariant all $H$-invariant $k$-relations). Consequently both, or neither, of $H$ and $H^{(k)}\\cap{\\rm\\Gamma L}(d,q)$ lie in ${\\mathcal C}$. As an application, we improve a 1992 result of Saxl and the fourth author concerning closures of affine primitive permutation groups.<\/jats:p>","DOI":"10.37236\/712","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:39:24Z","timestamp":1578695964000},"source":"Crossref","is-referenced-by-count":6,"title":["Invariant Relations and Aschbacher Classes of Finite Linear Groups"],"prefix":"10.37236","volume":"18","author":[{"given":"Jing","family":"Xu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Giudici","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cai Heng","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cheryl E.","family":"Praeger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,11,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p225\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p225\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T17:58:46Z","timestamp":1579283926000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p225"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,11,21]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/712","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,11,21]]},"article-number":"P225"}}