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Let $G$ be a solvable group in which all non-trivial subgroups are supplemented. (A subgroup $H$ of a group $G$ is called supplemented if $G=HK$ for some proper subgroup $K$. Such a subgroup $K$ is called a supplement of $H$ in $G$). Suppose further that $G$ has a maximal subgroup $M$ having only finitely many supplements in $G$.

Is $G$ necessarily finite?

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Dear Prof. Abdollahi, I am thinking about $Q_8 \times G$, where $G$ is a suitable infinite abelian poly-cyclic group. Did you examined such groups? –  Shahrooz Apr 27 '13 at 20:31
    
@Shahrooz: You mean if one takes $G=Q_8\times A$ for a suitable infinite finitely generated abelian group $A$, then $G$ may satisfy conditions given in my question? I think these such groups cannot be counterexample for my question, as the question has positive answer for supersolvable groups. Note that abelian polycyclic groups are exactly finitely generated abelian groups. –  Alireza Abdollahi Apr 28 '13 at 2:26
    
Dear Prof. Abdollahi, after some attempts, I could not find suitable $A$. –  Shahrooz Apr 28 '13 at 19:20

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