How do you show that there are no simple groups of order $2^n\times 5$ for $n\geq 4$, without using the theorem that a finite group of order $p^nq^m$, where p, q are primes and $m,n\geq 1$ is not simple.
I have a hint to 'use the coset action determined by the Sylow 2-subgroup', but I'm not sure what this means.