A characterization of Janko’s simple group

Abstract

If G is a finite group, we say that a series of subgroups G = Go > G > * > Gn= 1 is a maximal series of length n of G if Gi is a maximal subgroup of Gi1, 1 _ i < n. A subgroup H of G is called mth maximal in G if there exists at least one maximal series H= Gm< Gm.,-< <Go= G. Groups all of whose second, third and fourth maximal subgroups are invariant have been completely classified, see Janko [4], where the relevant results are enumerated. Further, those finite simple groups whose fifth maximal subgroups are trivial have been found by Janko [5]. Since Janko [6] has announced the discovery of a new simple group J, whose sixth maximal subgroups are all trivial, it may be of interest to classify those finite simple groups whose maximal chains are of length at most six. Thompson [8] has given the following DEFINITION. We say that a finite group G is an N-group if the normalizer of any nontrivial solvable subgroup is itself solvable. In the Main Theorem of [8], all simple N-groups are classified. These are the following groups: PSL(2, q), q a prime power greater than 3, PSL(3, 3), M,,, A7, Sz(22n+1) and PSU(3, 32). Since these groups have been studied elsewhere in great detail, and have been variously characterized group theoretically, we will consider these groups as known. Then we have the