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Helmut Wielandt discussed an old question (Chap. 2, Section 15, which can be dated back to Camille Jordan):

Let $g\neq 1$ be a permutation in some finite primitive permutation group $G$ of degree $n$. The minimal degree $m$ is defined to be the least number of points that $g$ permutes. It is known that if $m>3$, then $n$ is bounded by $$\frac{m^2}{4}\log\frac{m}{2}+m\left(\log\frac{m}{2}+\frac{3}{2}\right).$$

Question: Do we know a better upper bound today (with CFSG and O'Nan-Scott etc.)?

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2 Answers 2

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You seem to be aware of the answer to your own question, since you give the reference to the paper of Guralnick and Magaard, which classifies groups of minimal degree $\leq n/2$. Therefore $n \leq 2m$ with explicit exceptions list in the G--M paper. See also the previous paper of Liebeck and Saxl, which is easier and does $m < n/3$.

Even without CFSG and O'Nan--Scott, we know a sharp bound today. It's a result of Babai that $m \geq c \sqrt{n}$. Babai's proof is purely combinatorial and extends to primitive coherent configurations (e.g., strongly regular graphs), and it's close to sharp for the groups $S_m \wr S_2 \leq S_{m^2}$ and $S_m \leq \mathrm{Sym} (\binom{m}{2})$. An exactly sharp bound follows from recent work of Sun and Wilmes (https://arxiv.org/abs/1510.02195), which implies a classification of primitive coherent configurations where the minimal degree is $\leq n^{2/3} (\log n)^{-C}$.

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I know there is an accepted answer, but this might be helpful as well.

There is a very recent paper of Burness and Guralnick (see https://arxiv.org/abs/2112.03967), where the primitive groups with $m\ge 2n/3$ are classified (in Theorem 4). Of course, the results are CFSG-based.

The introduction of this paper contains very detailed history of bounding the minimal degrees and the fixed point ratios of finite primitive groups, which should be helpful.

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