Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

32

SUBSTITUTION GROUPS

[CH. II

Since similar remarks apply to every other row, it results
that X =n. That is, the order of the subgroup formed by all the
substitutions of a transitive group which omit a given letter is
equal to the order of the group divided by its degree. From the
given rectangle it follows that each letter occurs gi(n-1)
times in the substitutions of G. Hence these substitutions
involve gn(n- 1) =g(n- 1) letters. That is, the average number
of letters in the substitutions of a transitive group is equal to the
degree of the group diminished by unity.* In particular, the
average number of letters in all the possible substitutions
on n letters is n-1, and this is also the average number in all
the positive substitutions on these letters when n> 2.
13. Intransitive Substitution Groups. One of the simplest
examples of an intransitive substitution group may be obtained
by multiplying together transpositions on distinct sets of letters. For instance, the intransitive group generated by the
following three transpositions,
ab, cd, ef,
is of order eight and contains seven substitutions of order 2,
besides the identity. This is a special case of the elementary
theorem which affirms that h transpositions on h distinct pairs
of letters generate a group of order 2h and of degree 2h. This
group is intransitive when h> 1.
By multiplying all the substitutions of any transitive group
by all those of another transitive group, represented on a distinct set of letters, there results an intransitive group whose
order is the product of the orders of these two transitive groups,
and whose degree is the sum of their degrees. Hence it is
clear that it is possible to. construct an unlimited number of
different groups from any given group by representing the
group on distinct sets of letters and then multiplying the substitutions in every possible manner. Groups obtained in this
manner are sometimes called powers of the given group, an index
being used to indicate the number of times the group was used
*This interesting theorem was given explicitly for the first time by G.
Frobenius, Crelle, vol. 101 (1887), p. 287.