If $G$ has a finite maximal subgroup then what can I say about G ? Is it finite or infinite or finitly generated group ?
A group has a Finite maximal group
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abstract-algebra
group-theory
1 Answers
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A Tarski monster is infinite, but all of its proper, non-trivial subgroups have order $p$, for a fixed prime $p$. So all proper, non-trivial subgroups are maximal and finite.
However, a group $G$ with your conditions is definitely finitely generated, as it is generated by the finite set of elements of the given maximal subgroup, plus any single element outside it.