I have a question regarding a result of Bochert (1897: Math. Ann 49(1),133-144). This article contains a lower bound for the minimal degree m of a 2-transitive group of degree n ($m > n/3 - 2/3 \sqrt{n}$ if $n > 3$) which makes use of the estimate $m \ge n/4$ if $n \ne 25$, already published in 1892 (Math. Ann 40(2), 176-193). Even for native speakers Bochert's original proofs are hard to understand. I would be also grateful if somebody could provide me with some additional information on this subject. (By the way, I have found a least one article (Ito) from the 1950s which makes use of the bound and in Wielandt's "Finite Permutation Groups" it is quoted without proof, cf. Theorem 15.1).
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2$\begingroup$ A reference is Theorem 5.4A(iii) and exercise 5.4.1 in Dixon and Mortimers "Permutation Groups" I think. $\endgroup$– Kasper AndersenCommented Jun 12, 2019 at 23:56
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1$\begingroup$ For some more information, see for example Babai: "On the order of doubly transitive permutation groups" (1982) and Guralnick, Magaard: "On the Minimal Degree of a Primitive Permutation Group" (1998) and the bibliography in these papers. $\endgroup$– Mikko KorhonenCommented Jun 13, 2019 at 9:15
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You should probably look at the first two of W.A. Manning's papers entitled The degree and class of multiply transitive groups.
In these papers, Manning proves variants on Bochert's results:
- In the first, he proves a stronger result with the extra assumption that there is an element of minimum degree of even order;
- In the second, he proves stronger results for the remaining cases -- where there are elements of minimum degree of odd prime order (see Theorem IV of the second paper).
Two important notes however:
In the second paper, he notes (on p. 644) that there is an error in Bochert's proof, involving an ascending series of limits;
In the second paper, he also notes that for the particular case where the elements of minimum degree are OF ORDER 3, then he does not quite achieve Bochert's result, but must decrease it by one. I am not aware, off the top of my head, if this lacuna in Bochert's result has ever been rectified (without CFSG).
You can find the two papers in question here: Paper 1 | Paper 2
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$\begingroup$ Thanks for your answer; I was aware of Manning's papers, but have to admit that I could not follow his arguments. Is there any modern account of Manning's result (without CFSG)? $\endgroup$ Commented Jun 13, 2019 at 10:48
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1$\begingroup$ I'm not aware of a direct modern treatment. However I think the key result of Bochert that you probably need t is the lemma stated by Manning at the start of the second paper. My impression is that you can prove this lemma using the method given by Wielandt (Theorem 14.2 of Finite Permutation Groups). The key thing is to show that if your permutation group contains two elements $a$ and $b$ that move a "small" number of common points, then the commutator $[a,b]$ will fix more points than either of $a$ and $b$. if you can figure this proof out, then Manning's papers give a complete argument. $\endgroup$ Commented Jun 13, 2019 at 11:10