Multiply Transitive Substitution Groups*

A substitution group G of degree n is said to be r-fold transitive if each of the n(n — 1) • • • (n—r + 1) permutations of its n letters taken r at a time is represented by at least one substitution of G. It is not sufficient to say that each of its possible sets of r letters is transformed into every one of these sets by the substitutions of G. For instance, in each of the semi-metacyclic groups of degree p, p being a prime number of the form 4«+3, every possible pair of its letters is transformed into every other such pair by the substitutions of the group and yet these groups are only simply transitive. Another well known definition of an r-fold transitive group of degree « is that this group contains a transitive subgroup of each of the degrees n, n —1, • -, n—r+1. It is obvious that these two definitions are equivalent and that if a group is r-fold transitive it is also (r—a)-fold transitive, where a is a positive integer g r — 1. It is not difficult to prove that a necessary and sufficient condition that a primitive group of degree n be r-fold transitive, r>l, is that it contain a doubly transitive group of degree n—r+2. This may have to coincide with the group itself if the group is only doubly transitive. To prove that such a group G must contain a triply transitive group of degree n — r+3, whenever r > 2 it is only necessary to note that G involves a substitution which transforms at least one letter of the doubly transitive group of degree » —r+2 into a letter of this group while it transforms another letter of this group into a letter not involved therein since G is primitive. Hence G involves at least two primitive subgroups of degree » — r+2 which have at least one common letter but do not have all their letters in common. It must therefore involve two such primitive groups which have all except one letter of each in common. These two groups obviously generate a triply transitive group since they themselves are doubly transitive. When r>3 we may repeat this argument and prove the existence in G of a four-fold transitive group of degree n — r+4, etc. It must therefore contain a transitive group of each of the degrees »,» — !,•••, n—r+1.