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Following [Neumann P.M., The lawlessness of groups of finitary permutation groups, Arch. Math. 55(6) 1990, 521-532], we define totally imprimitive groups in a more general form as follows:

Let $G$ be a transitive subgroup of Sym$(\Omega)$ for some countably infinite set $\Omega$. If there is a chain of proper $G$-blocks $$\Delta_0\subset \Delta_1\dots\subset \Delta_i\subset\dots$$ such that $\bigcup_{i\in\omega}\Delta_i=\Omega$, i.e. there is no maximal $G$-block, then $G$ is called totally imprimitive. My question is:

Is there a totally imprimitive subgroup $G$ such that Alt$(\Omega)<G< $Sym$(\Omega)$, where Alt$(\Omega)$ is the alternating subgroup of Sym$(\Omega)$?

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    $\begingroup$ I'm confused by your "i.e. $\Omega$ has no maximal $G$-block. If $\Omega=G/H$, the condition seems to be that there is an increasing sequence of proper subgroups $H_n$ with $H_0=H$ and $\bigcup H_n=G$. But this might happen while there is also a maximal proper subgroup $L<G$ containing $H$. Therefore, I don't understand what the definition of "totally imprimitive" is. $\endgroup$
    – YCor
    Commented Jun 25, 2023 at 10:39
  • $\begingroup$ Is there any problem in Neumann's definition in the cited paper? Neumann separates the imprimitivity condition as ``almost imprimitive" and "totally imprimitive". $\endgroup$
    – IGT
    Commented Jun 25, 2023 at 10:59
  • $\begingroup$ But surely ${\rm Alt}(\Omega)$, and hence also all groups containing it, is $2$-transitive, and therefore primitive. Or have I misunderstood something? $\endgroup$
    – Derek Holt
    Commented Jun 25, 2023 at 11:24
  • $\begingroup$ Looking at the paper, there is no problem in the paper. But it seems that there is problem in your question: P. Neumann never writes that the existence of such a sequence implies totally imprimitive. In addition, in this paper he mostly assumes that the group acts by finitary permutations, which is a huge assumption. $\endgroup$
    – YCor
    Commented Jun 25, 2023 at 11:25
  • $\begingroup$ In [Cansu Betin&M. Kuzucuoglu, On locally graded barely transitive groups, Cent. Eur. J. Math. 11(7) (2013), 1188-1196] uses the same definition. Is it meaningful? $\endgroup$
    – IGT
    Commented Jun 25, 2023 at 11:35

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