# is there group such that has finite maximal subgroup and index of all proper subgroup of $G$ in $G$ be infinite?

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 ($\forall H\lt_{s.g}G);H\neq G$ : [$G:H$]=$\infty$)

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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... – DonAntonio Apr 16 '13 at 16:34

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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– Andreas Caranti Apr 16 '13 at 18:00