Can a group of order 3000 be a simple group? Can a group of order 3000 be a simple group? How about the case of a group of order 1000?
A group of order 1000 has a normal Sylow 5-subgroup, so it cannot be simple.

A group of order 3000 has either a normal Sylow 5-subgroup or 6 of them. If there are 6 Sylow 5-subgroups, then the normalizer of a Sylow 5-subgroup has order 3000/6 = 500. A subgroup of order 500 cannot have a trivial normal core in a group of order 3000: Index of the normal core in the Sylow 5-subgroup must divide (6 - 1)! but 500 does not divide 120 = 5!.
let $G$ be a group of 3000 .Then by SYLOWS theorem $G$ contain a subgroup $H$ of order 1000. as$ 0(G)$ does not devide factorial of $(i(H))$. $H$ contain a non trivial normal subgroup $K$ of $G$ which contain in $H$. so $G$ is not simple. where $i(H)$ means index set of $H$ in $G$