Solvable transitive groups of prime degree

This is exercise 7.2.12 of Robinson's Course in the Theory of Groups, page 195 in the first edition.

A transitive subgroup of prime degree is primitive, and primitive solvable groups have a regular normal subgroup that is complemented by a unique conjugacy class of maximal subgroups. In particular, the Sylow p-subgroup C of order p is that regular normal subgroup, and the complement (being a permutation group) acts faithfully on it. In other words, the centralizer of the subgroup C is C itself. Hence G/C is a subgroup of Aut(C), so cyclic of order dividing p-1.

Since G is solvable it has a Sylow p-complement M, and by Hall's 1928 theorem, the number of
such Sylow p-complements is a divisor of p. If it is 1, then M is normal, so M centralizes C, so M=1, and G=C is cyclic.

This is summarized by saying that G is a subgroup of AGL(1,p) containing the translation subgroup. More generally every primitive solvable group is a subgroup of AGL(n,p) where p^n is the degree of the permutation action (but AGL(n,p) is no longer solvable itself).