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its degree exceeds 1,000, while the formula \n + 1 would place the upper limit of transitivity for such groups beyond 300. These illustrations may suffice to exhibit clearly that a much smaller upper limit for the degree of transitivity of a primitive group which is neither alternating nor symmetric results from the use of the present theorem than the one given by \n + 1, whenever n is large. When n = 12 = 7 + 5 the two theorems lead to the same upper limit. This is also true for the cases when n is 8 or 9. Since the groups whose degrees are less than 8 are so well known, it does not appear necessary to preserve the formula \n + 1 as an upper limit of the degree of transitivity of substitution groups which do not include the alternating group, especially since the theorem proved above is based upon such very elementary considerations.

The permutations of the natural numbers can not be well ordered