Invariant Relations and Aschbacher Classes of Finite Linear Groups

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.

[1]  D. E. Taylor.  The geometry of the classical groups , 1992 .

[2]  Gerhard Hiss,et al.  Low-Dimensional Representations of Quasi-Simple Groups , 2001, LMS J. Comput. Math..

[3]  Gunter Malle,et al.  CORRIGENDA: LOW-DIMENSIONAL REPRESENTATIONS OF QUASI-SIMPLE GROUPS , 2002 .

[4]  B. Huppert.  Endliche Gruppen I , 1967 .

[5]  On subgroups of the special linear group containing the special orthogonal group , 1985 .

[6]  Oliver H. King,et al.  The subgroup structure of finite classical groups in terms of geometric configurations , 2005, BCC.

[7]  Jing Xu,et al.  On closures of finite permutation groups , 2006, Bulletin of the Australian Mathematical Society.

[8]  Martin W. Liebeck,et al.  The Subgroup Structure of the Finite Classical Groups , 1990 .

[9]  R. Carter.  THE SUBGROUP STRUCTURE OF THE FINITE CLASSICAL GROUPS (London Mathematical Society Lecture Note Series 129) , 1991 .

[10]  Michael Aschbacher,et al.  On the maximal subgroups of the finite classical groups , 1984 .

[11]  K. Zsigmondy,et al.  Zur Theorie der Potenzreste , 1892 .

[12]  H. Wielandt,et al.  Permutation groups through invariant relations and invariant functions , 1969 .

[13]  Peter J. Cameron,et al.  2-Transitive and antiflag transitive collineation groups of finite projective spaces , 1979 .

[14]  L. Kovács.  Primitive Subgroups of Wreath Products in Product Action , 1989 .

[15]  J. Dixon,et al.  Permutation Groups , 1996 .

[16]  Closures of Finite Primitive Permutation Groups , 1992 .

[17]  Ernst Snapper,et al.  Metric affine geometry , 1973 .

[18]  A note on arc-transitive circulant digraphs , 2008 .

[19]  Frank Lübeck.  Small Degree Representations of Finite Chevalley Groups in Defining Characteristic , 2001, LMS J. Comput. Math..

[20]  G. Miller.  Linear Groups with an Exposition of the Galois Field Theory , 1902 .

[21]  Michio Suzuki.  Group Theory I , 1981 .

[22]  J. Conway,et al.  ATLAS of Finite Groups , 1985 .

[23]  On subgroups of the special linear group containing the special unitary group , 1985 .