Density of Sylow subgroups

Question

Let G be any group of order n.Also assume p be the largest prime dividing n.Let n(p) be the maximum no of sylow subgroups a group of order n can have.Is it possible to sa anything definite about the limit n(p)/n as n tends to infinity?

Answer 1

The limit does not exist, but it is easy to see that there are subsequences of $1,2,3,\ldots$ (such as prime powers) on which $n(p)/n=1/n$, which tends to $0$, and other subsequences (such as $n=2^{2k}3$) with $n(p)/n=1/3$, which I think is the largest possible.