# Minimal degree of primitive permutation group

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.)?

• A comment probably useful to others: See the paper of Guralnick and Magaard here: doi.org/10.1006/jabr.1998.7451. Mar 25, 2022 at 8:23

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 $$\leq n^{2/3} (\log n)^{-C}$$.
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.