On a class of doubly transitive groups

THE class u(u> 3) of a doubly transitive group of degree n is, according to Bochert,f greater than \n — § Vw. If we confine our attention however to those doubly transitive groups in which one of the substitutions of lowest degree is of order 2, it appears that the class is greater than \n — \ Vw — 1. The proof of this statement rests essentially upon the following LEMMA. The degree of a diedral group of class u generated by two non-commutative substitutions of order 2 and degree u is at most §w. Let s and t be the two substitutions in question, and let the order of their product be N = 2pip2 a2 • • • Pn, where pi> P2, ' are distinct odd primes. The transitive constituents of {s, t] may be arranged as follows: s has mi cycles displacing letters not in t, and t has ra2 cycles displacing letters not in s; there are Xi regular constituents of order X{, with a generator in both s and t (thus common cycles of s and t are explicitly included, while the preceding type of constituent of degree and order 2 is excluded) ; there are yj non-regular constituents of degree Yj and order 2Yj, Yj an odd number; there are yd non-regular constituents of degree Yd and order 2 Yd, Yd even, with the generator of degree Yd in s, and the generator of degree Yd — 2 in t; in like manner there are yd' constituents of the order Yd with Yd — 2 letters in s and Yd letters in t. Since transitive