TRANSITIVE PERMUTATION GROUPS OF DEGREE p = 2q+l,p AND q BEING PRIME NUMBERS

1. Introduction. Let p be a prime number such that q — l{p — l) is also a prime. Let Q, be the set of symbols 1, • • • , p, and ® be a non-solvable transitive permutation group on 12. Such permutation groups were first considered by Galois in 1832 [I, §327; III, §262]: if the linear fractional group LF 2 (l) over the field of I elements, where I is a prime number not smaller than five, contains a subgroup of index I, then I equals either five or seven or eleven. These three permutation groups will be denoted by A$, GV and Gu. G7 has degree 7 and order 168; Gu has degree 11 and order 660. Next in 1861 two permutation groups, one, which has degree 11 and order 7,920, and the other, which has degree 23 and order 10,200,960, were found by Mathieu [l6; 17]. These two permutation groups will be denoted by Mn and