Is a group with order $16*17$ solvable?
I know that from Burnside this is solvable since 2 and 17 are prime and 4 is greater than 0. However, I am not allowed to use it, so what should I do?
Thanks in advance
I noticed that there is a similar thread Group of order $8p$ is solvable, for any prime $p$
Number of Sylow-$17$ subgroup is congruent to $1$ mod $17$, and it divides $2^4$, hence it must be $1$; so Sylow-$17$ subgroup is unique, hence normal, call it $P$. Then $G/P$ is group of order $2^4$; since $p$-groups are solvable, it follows that both $P$ and $G/P$ are solvable, hence $G$ is solvable.