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I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and Wolf", but I need to know which of them hold in the unsoluble case.

More precisely, I arrive at the the action of the group $G$ which is faithful and quasi-primitive and every stabilizer in this action contains a $p$-Sylow subgroup, where $p\in\pi(G)$. I know if $G$ is solvable, then what happens for $G$ but for me, $G$ is an arbitrary group

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  • $\begingroup$ This paper is probably a good place to start: researchgate.net/publication/… $\endgroup$
    – Colin Reid
    Commented Jul 25, 2014 at 13:39
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    $\begingroup$ Your question needs to be more specific, I think. You need to tell us more precisely what you want/need to know $\endgroup$
    – Geoff Robinson
    Commented Jul 25, 2014 at 14:50
  • $\begingroup$ Dear Professor Robinson, I arrive at the the action of the group $G$ which is faithful and quasi-primitive and every stabilizer in this action contains a $p$-Sylow subgroup, where $p \in \pi(G)$. I know if $G$ is solvable, then what happens for $G$ but for me, $G$ is an arbitrary group. $\endgroup$
    – sara
    Commented Jul 25, 2014 at 15:25
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    $\begingroup$ @Colin Reid: I believe that the OP means the notion of quasi-primitivity for linear groups, which is quite different from the one for permutation groups. $\endgroup$
    – Peter Mueller
    Commented Jul 25, 2014 at 18:32
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    $\begingroup$ @Peter Mueller: Let $r$ be a prime and $G \leq GL(d,r)$ haveing order divisible by the prime $p \neq r$ such that every orbit length on vectors is coprime to $p$. If $G$ is solvable, then the structure of $G$ is known for me. What happens for $G$ in the unsovable case? $\endgroup$
    – user56484
    Commented Jul 27, 2014 at 7:00

3 Answers 3

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This preprint by Giudici, Liebeck, Praeger, Saxl, and Tiep might be useful to you. They classify all subgroups $G$of $GL(d,p)$ having order divisible by $p$ but every orbit length on vectors coprime to $p$.

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  • $\begingroup$ Thank you very much for answer, but actually in my case $G \leq GL(d,r)$ such that $r \neq p$. $\endgroup$
    – sara
    Commented Jul 26, 2014 at 17:43
  • $\begingroup$ Earlier comment deleted:I had not noticed the characteristic restriction. $\endgroup$
    – Geoff Robinson
    Commented Jul 26, 2014 at 19:41
  • $\begingroup$ @sara: I think you should make your question clearer. $\endgroup$
    – Peter Mueller
    Commented Jul 26, 2014 at 20:07
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You might find useful the paper of Carlo Casolo, "Some linear actions of finite groups with $q^\prime$-orbits". If I understand correctly, this handles nonsolvable groups under the additional assumption that the stabilizer of each vector includes a Sylow subgroup of $G$ as a central subgroup. If you have access to MathSciNet, you can find the paper at

http://www.ams.org/mathscinet-getitem?mr=2661653

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    $\begingroup$ John, that link looks wustl.edu specific. Try ams.org/mathscinet-getitem?mr=2661653 instead. (And no mathscinet access is required for that link.) $\endgroup$
    – Russ Woodroofe
    Commented Jul 28, 2014 at 2:46
  • $\begingroup$ @John Shareshian and @ Russ Woodroofe, thank you very much for your answers. $\endgroup$
    – sara
    Commented Jul 30, 2014 at 12:27
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Your question is rather vague, but the more specific information you provide suggests that you have a linear action of a group on a (presumably finite) vector space in which all orbits on vectors have length prime to $p.$ You have presumably reduced this to the case where the linear action is quasiprimitive. It is "generically" the case that when a finite group $G$ acts faithfully as a group of linear transformations on a vector space, there is a vector which is stabilized by the identity, but nothing more (ie, there is a regular orbit on vectors). However, there are exceptions to this, some well understood, some not so well. Much work has been done on this in the case of solvable groups, and the book of Manz and Wolf that you mention contains much information of the state of that problem for solvable groups prior to its publication date. People such as Liebeck, Saxl and Guralnick have studied when quasisimple groups and close relatives have regular orbits on vectors when they act faithfully as liner groups. The generalized Fitting subgroup of a finite group $G$ which has a faithful quasiprimitive linear action is rather restricted. It is (when the field is large enough) a central product of a class $2$-nilpotent group (the usual Fitting subgroup), and some quasisimple subnormal subgroups. Questions about regular orbits in such actions (and other orbit questions) often reduce to the case where this generalized Fitting subgroup is either extraspecial (or very close to it), or else quasisimple. When the generalized Fitting subgroup is extraspecial, the analysis is quite similar to the solvable case (but somewhat more complicated). When the generalized Fitting subgroup is quasisimple, the results of simple group specialists come into play.

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