...half the degree of the group. In particular, if in a transitive substitution group G the subgroup Gi composed of all the substitutions of G which omit a given letter is transitive and omits half the letters of G, then the number of systems of imprimitivity of G is...

...begin with the case where the substitution group G is transitive and of degree n. If the subgroup Gl composed of all the substitutions of G which omit a given letter is of degree n — 1, there is no substitution which involves any of the letters contained in the substitutions...

...substitutions on these n letters, in addition to the identity. The order of K is a, the degree of the subgroup composed of all the substitutions of G which omit a given letter being n — a. Hence a necessary and sufficient condition that K be transitive is that G be regular,...

...G is thus represented, it involves n conjugate subgroups Gì, G£, . • • , Gn each of which is composed of all the substitutions of G which omit a given letter. If the subgroup of Gì, composed of all its substitutions which omit a given letter, is of degree n...

...that when G is thus represented, it involves n conjugate subgroups Gi, G2, . . . , Gn each of which is composed of all the substitutions of G which omit a given letter. If the subgroup of Gi, composed of all its substitutions which omit a given letter, is of degree n—...

...G which are such that each system is composed of two letters. When G is non-regular the subgroup GI composed of all the substitutions of G which omit a given letter a^ must omit all the letters of the system including at in the various possible set of systems which are...

...automorphism in question. It has therefore been proved that every automorphism of G in which the subgroups composed of all the substitutions of G which omit a given letter correspond to such subgroups is effected by substitutions of the largest group on the letters of G...

...automorphism in question. It has therefore been proved that every automorphism of G in which the subgroups composed of all the substitutions of G which omit a given letter correspond to such subgroups is effected by substitutions of the largest group on the letters of G...

...subgroups. The third category of automorphisms of G is composed of all those in which the subgroups composed of all the substitutions of G which omit a given letter correspond to subgroups of degree n. Such automorphisms are not always possible. In fact it may happen...

...substitutions on these M letters, in addition to the identity. The order of K is a, the degree of the subgroup composed of all the substitutions of G which omit a given letter being n — a. Hence a necessary and sufficient condition that K be transitive is that G be regular,...