On an Undemonstrated Theorem of the Disquisitiones Arithmetics

than n — 1, while the average number of elements in all the substitutions of G is n — 1. This is impossible, as the substitutions of G which are not also in H cannot contain more than n — 1 elements. Hence H is transitive. Since the average number of elements in the substitutions of G is the same as that in the substitutions of H, the substitutions of G which are not found in H must all be of the (n — ï)th degree. The theorems of §§ 75, 76 of Getto's work * are proved by the preceding paragraph. It may be well to add in regard to the given § 75 that G may be transitive while the corresponding subgroup of G is intransitive. The following group | is an instance :