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give an example of group like $G$ that satisfied :

  • $G$ has finite maximal subgroup

  • index of all proper subgroup of $G$ in $G$ be infinite ($\forall H\lt_{s.g}G);H\neq G $ : [$G:H$]=$\infty$)

Thanks in advance

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  • $\begingroup$ Those two conditions seem to contradict each other: the second one means $\,G\,$ is infinite, yet the first condition wants a maximal subgroup which is finite...! If it is finite then its index is infinite and the 2nd condition cannot be fulfilled... $\endgroup$ – DonAntonio Apr 16 '13 at 16:34
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I can answer this affirmatively by reference. Wikipedia page on the Burnside problem says:

In 1982 Ol'shanskii was able to strengthen his results to establish existence, for any sufficiently large prime number $p$ (one can take $p > 10^{75}$) of a finitely generated infinite group in which every nontrivial proper subgroup is a cyclic group of order $p$.

Now, in such a group $G$ as Ol'shanskii has constructed, every nontrivial proper subgroup is maximal, finite, and has infinite index. Unfortunately, I don't know Ol'shanskii's construction itself.

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