Based on the answer to this question, I am wondering:
Let $n>8$ and $G$ a finite group for which all orders of its elements are contained in $\{1,\dotsc,n\}$ (denote the set of all such groups by $\mathcal G_n$). How big can $\dfrac{|\mathcal O(G)|}{n}$ be as $n$ grows? Here $\mathcal O(G)$ denotes the set of all orders of elements of $G$.
From the previous question, we know $f(n):=\max\limits_{G\in \mathcal G_n} \dfrac{|\mathcal O(G)|}{n}<1$ for $n>8$.
- Is it clear that $\lim\limits_{n\to\infty}f(n)=0$? If so, are there any estimates for the decay?
- Are there specific values of $n>8$ for which "good" lower bounds are known, i.e. groups with a relatively high density of $\mathcal O(G)$ ?
