On multiply transitive groups
W. A. Manning · Transactions of the American Mathematical Society · 1906
The properties of primitive groups that contain transitive subgroups of lower degree were first investigated by JORDAN.t In the Traite des Substitutions he proved that if a pritnitive group of degree nt contains a circular substitution of prime degree (pv) it is at least n p + 1 times The capital importance of this theorem led him to examine the general and much more difficult case, that in which the subgroup of lower degree is merely transiti ve. He obtained the remarkable theorem: t ' If a primitive group G of degree n contains a group F, the substitutions of which displace only 1letters and permute them transitively (p being any integer), it is at least n -p 2q + 3 times transitive, q being the greatest divisor of p such that we can arrange the letters of F in two different ways in systems of q letters which have the property that each substitution of r replaces the letters of each system by those of a single system. If none of the divisors of p have this property (which will happen notably if F is primitive, or formed of the powers of the same circular substitution) G is n p + 1 times transitive. NETTO ? and RUDIO 11 later gave proofs for that special case in which F is primitive. The only other contribution to this theory was made by MARGGRAFF.? He proved that if q is the greatest divisor of p such that the letters of r may be arranged in systems of imprimitivity in at least three different ways, and if p is divisible by some number r such that F admits r + 1 systems of imprimitivity with one letter in common and no two of which have more than one letter in common, G is at least n p 2q + 3 times If these two conditions are not fulfilled, G is n p + 1 times In MARGGRAFF'S