Density of Sylow subgroups

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?

• Choosing $\;n=p\,,\,p^2\,,\,p^3\,,\ldots\;$ , we get a subsequence of the above that converges to zero, so if the limit exists it is zero. Oct 30, 2014 at 19:02
• Did you look at symmetric groups?
– Pedro
Oct 30, 2014 at 20:35

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.