A theorem of Marshall Hall in Theory of Groups

Question

In his text M Hall recites a theorem that says a permutation group of degree $n=kp+r$, with $p>k$, $r>k$, and $n\ne p+2$, that is $(r+1)$-fold transitive must be $S_n$ or $A_n$. (Theorem 5.7.2.) The proof is complicated and I think it is written to readers who know it already. I am seeking a second source to help me understand the argument. At one point Hall writes, "therefore the only factor group isomorphic with a factor group of $A_r$ (the alternating group on $r$ letters) is the identity." The only factor group of $A_r$ is the identity or $A_r$ itself, as it has no proper normal subgroups ($r>4$ here). I find this line distracting, but I assume I am missing something. The rest of the proof is similarly not perspicuous. I would love to see the homomorphism alluded to sketched, for example, or even fleshed out in detail! Any leads would be appreciated. Hall provides no references for this theorem, but I assume it is standard among experts.

Answer 1

That particular statement you find distracting is just a consequence of the fact that $A_n$ is a simple group for $n>4$, which is indeed a standard result, and which Marshall Hall has a few pages previously as Theorem 5.4.3.

As for other references for the result stated as Theorem 5.7.2 (which you slightly misstated...it also requires $p>k$, $r>k$, and $n\ne p+2$), I don't know that there are any other than Miller's original paper (see below). Passman's and Dixon & Mortimer's books on permutation groups don't give this specific theorem. However, Chapter 21 of Passman and more so Chapter 7 of Dixon and Mortimer do have extensive discussions of similar theorems. On the other hand, Hall's 5.7.2 is the sort of proof it pays to study. You may find it difficult right now, but compared to the proof of the structure theorem for non-solvable Frobenius complements, to give one example from Passman's book, it's pretty straightforward.

While Hall does not explicitly so state, it appears that 5.7.2 is a 1915 result due to the same Miller whom he credits with 5.7.1. See https://projecteuclid.org/download/pdf_1/euclid.bams/1183423390 . I haven't read through it in detail, but I suspect reading it will give you a greater appreciation for Hall's version (and also a sense of how mathematical prose style has changed from Miller in 1915 to Hall in 1959 to today).