# What is an upper limit of relative size of conjugacy class of the transitive finite group?

What is

$$\limsup_{n\to\infty} \sup_{\deg(G)=n} \left( \frac{\max_{x\in G} \left|\operatorname{conj}(x)\right|}{\operatorname{ord}(G)}\right),$$

$$G$$ transitive permutation group?

And what are the examples of finite permutation groups containing a large conjugacy class?

$$\deg(G)$$ resp. $$\operatorname{ord}(G)$$ are degree and order of $$G$$, $$\operatorname{conj}(x)$$ is the conjugacy class of $$x\in G$$

• Whenever $n$ is odd, the dihedral group of order $2n$ acts transitively and has a conjugacy class (the order two elements) of order $n$. So $1/2$ is achievable. – David E Speyer Oct 17 '18 at 15:57
• You mean "acts transitively and faithfully on $n$ elements". – YCor Oct 17 '18 at 16:48
• So well, clearly $|conj(x)|\le |G|/2$ for every nontrivial finite group and $x\in G$, so the term in parenthesis is $\le 1/2$ for every $G$ transitive on $n\ge 2$ elements. So David's remark settles the problem: the limsup is 1/2. – YCor Oct 17 '18 at 16:51