ON SUBGROUPS OF M24. II: THE MAXIMAL SUBGROUPS OF M2i

In this paper we effect a systematic study of transitive subgroups of M24, obtaining 5 transitive maximal subgroups of M24 of which one is primitive and four imprimitive. These results, along with the results of the paper, On subgroups of M2i. I, enable us to enumerate all the maximal subgroups of M24. There are, up to conjugacy, nine of them. The complete list includes one more in addition to those listed by J. A. Todd in his recent work on M24. The two works were done independently employing completely different methods. In this paper we effect a systematic study of transitive subgroups of M24, obtaining five transitive maximal subgroups of M2i. This result, along with the previous results on the maximal subgroups among the intransitives [3], enables us to enumerate all the maximal subgroups of M24. There are, up to conjugacy, nine of them. We dispose of the study of primitive subgroups by observing that a proper transitive subgroup of M2i is either PSL2 (23) or imprimitive (Proposition 1.1). Six different types of systems of imprimitivity can be obtained from the 24 points of M2l, Ü, viz., 24/n blocks of length n for n = 12, 8, 6, 4, 3, and 2, respectively. The systems of imprimitivity with 24/n blocks of length n are denoted by n'|n'|... |nf. Obviously, when n 3:6, the systems are of the same type as sets of n distinct points, as defined in the preceding paper [3, p. 1]. An imprimitive group G with the above type of systems of imprimitivity will be called an imprimitive group of type nm where n • m = 24. The kernel of imprimitivity, viz., the normal intransitive subgroup of G which contains all the substitutions which do not interchange the systems of imprimitivity n'|n'|... |nf, is usually denoted by K. Let ß, denote a system of imprimitivity. Then, in general, K is constructed by multiplying the elements of certain cosets of the constituents KBi which correspond by an isomorphism. Let K¡ denote the kernel of the restriction of the kernel K on B,. If Kf is the identity, then K is built by establishing a simple isomorphism between the corresponding substitutions of KBi. In this case we denote KxKB^\KBA ... \KBt. The image of imprimitive group G will be denoted by G*. If the transitive group Received by the editors August 3, 1970. AMS 1969 subject classifications. Primary 2020, 2029.