Transitive permutation groups of degree 𝑝=2𝑞+1, 𝑝 and 𝑞 being prime numbers. II
Noboru Ιτο · Transactions of the American Mathematical Society · 1964
Introduction.Let p be a prime number such that q = ^(p -1) is also a prime number.Let Q be the set of symbols l,--,p, and (5 be anonsolvable transitive permutation group on O.In [10] and [11] the structure of such a permutation group © has been studied, and, in particular, the following theorem [11, Theorem 1] has been proved: If r = \(p-3) is also a prime number, then © is triply transitive.The purpose of this work is to remove this additional assumption, namely, to prove the following theorem.Theorem.If © is not triply transitive, then © is isomorphic to either LF(2,1) with p = l or LF(2,1T) with p = 11.Hence, in particular, if p> 11, then © is triply transitive.Now let yi be a minimal normal subgroup of ©.Since © is obviously primitive, 91 is transitive on fi.Let 5ß be a Sylow p-subgroup of ©.Then 5ß is contained in 51.As a minimal normal subgroup 9t is a direct product of mutually isomorphic simple groups.Since the order of 91 is divisible by p only to the first power, 91 must be simple.On the other hand, by a theorem of Sylow, we have that © = 9iAs5ß, where As*B denotes the normalizer of ^ß in ©.Since 5ß obviously coincides with its own centralizer in ©, As5ß/5ß is a cyclic group of order dividing p -1.Since we have that © /9t = As5ß /As5ß n 9t and As^ßn5i 2'iß, ©/9Î is also a cyclic group of order dividing p-1.Since © is nonsolvable, 91 is nonsolvable, too.Therefore in order to prove the theorem we can assume the simplicity of ©.So from now on let © be simple.If As5ß = 5ß, then by a splitting theorem of Burnside © contains a normal Sylow p-complement.Since © is simple, this implies that © = 5ß, contradicting the nonsolvability of ©.If the order of As5ß is even, let us consider an involution in As^ß.The cycle structure of this involution consists of q transpositions, and it is odd, contradicting the simplicity of ©, because we can obviously assume the oddness of q.Hence the order of As5ß must be equal to qp.Let Qbe a Sylow g-subgroup of As5ß such that G fixes the symbol 1 of Q.If Q is not a Sylow g-subgroup of ©, then © contains a g-cycle.Hence by a theorem of Jordan [17] © coincides with the alternating group 91 on Q. 51 is triply transitive for p ^ 5. Therefore in order to prove the theorem we can assume that JQ is Received by the editors