Let $G$ be a finite solvable group with $|G|=p_{1}^{\alpha _{1}}\cdot \cdot\cdot p_{r}^{\alpha _{r}}$ such that $p_{1}$,..., $p_{r}$ are distinct primes. By Hall's theorem, we have: If the number of Sylow $p_{i}$-subgroups is $q_{1}^{\beta _{1}}\cdot \cdot \cdot q_{s}^{\beta _{s}}$, then for all $i\in \{1,2,...,s\}$ , $q_{i}^{\beta _{i}}\equiv 1$ (mod $p_{i}$).
My question: Is the converse true? That is, if for every $p_{i}$ the number of Sylow $p_{i}$-subgroups is $q_{1}^{\beta_{1}}\cdot \cdot \cdot q_{s}^{\beta _{s}}$, and for all $i\in \{1,2,...,s\}$ , $q_{i}^{\beta _{i}}\equiv 1$ (mod $p_{i}$), then $G$ is solvable?
I have checked it for all the small nonsolvable groups and many others. I discovered that except for $PSL(2,7)$ and groups that have $PSL(2,7)$ as the composition factor, the rest do not have this property.
We must first answer the following question:
For a simple group $S\neq PSL(2,7)$ why there exists prime $p$ such that if the number of Sylow $p$-subgroups of $S$ is $q_{1}^{\beta _{1}}q_{2}^{\beta_{2}}\cdot \cdot \cdot q_{t}^{\beta _{t}}$, then at least for some $1\leq i\leq t$, $q_{i}^{\beta _{i}}\ncong 1$ (mod $p$).
Can anyone help in solving the above question?
