Vol. 8, 1922 
MATHEMATICS: G. A. MILLER 
239 
the average number of letters, viz. n — x — \, the number of substitu- 
tions of 6^ which omit at least one letter is ngi {% -\- \) = g {% -\- 1) 
gi being the order of Gi. This case evidently presents itself only when G 
is regular. In all other cases this number of substitutions must be greater 
since each such case can be constructed by taking successively one letter 
away from some one substitution in the idealized Gi' in which all the sub- 
stitutions including the identity are of the same degree, and adding it to 
another. By each such operation the number of substitutions which omit 
at least one letter is increased, since the number of such substitutions is 
thereby first diminished by = , where ^ > x + 1, 
k k ^ \ k{k ^ \) 
71 71 71 
and then increased by — 1 = where I > x 1. As 
/ - 1 / /(/ - 1) 
each such operation increases the total number of substitutions which 
omit at least one letter this number must be increased by the totality of 
such operations which lead from the idealized G\ to the real Gi. It 
should be noted that the increase due to one such operation may be a 
fractional substitution. 
In the special case when G is multiply transitive Gi must be transitive 
and of degree n — 1. In this case g/{x + 1) = g/2 and by reducing one 
substitution of G'l to the identity we increase the number of substitutions 
whose degree is less than n by 
fl{7l ~ 2) _ W _j_ ^ 
2 2 
If Gi contains substitutions whose degree is less than n — 2 the number of 
substitutions of degree 71 — 1 must be still larger. Hence it results directly 
that in a multiply transitive substitution group of degree n the number of 
substitutions of degrees n — 1 and n — 2 cannot be less than 
g .n{7t - 3) 
2-^— 
and this is the exact number of such substitutions when G involves no 
substitution besides the identity whose degree is less than n — 2 and only 
then. 
The main theorem of this note may be stated as follows: I71 every non- 
regular transitive substitution group of degree n the number of the substitu- 
tions which omit at least one letter exceeds g/x where x is the number of the 
transitive constituents in a subgroup composed of all the substitutions which 
omit one letter regarded as a group of degree n. One reason for emphasizing 
this matter here is that the theorem is stated incorrectly even for the 
special case when the group is primitive on page 357 of Cajori's History of 
