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

38

SUBSTITUTION GROUPS

[Cn. II

EXERCISES
1. Every transitive group of degree n involves at least n-1 substitutions which separately involve all the letters of the group; if it contains
also substitutions of degree n-a, I <a <n, it must contain more than
n-1 substitutions of degree n.
Suggestion. The average number of letters in the substitutions is n-1.
2. If all the substitutions which omit a given letter of a transitive
group of degree nZ constitute a group of degree in-1, then the ni conjugates
of the latter are transformed under the transitive group in exactly the same
way as its letters are transformed.
3. A transitive group composed of invariant substitutions is necessarily
regular.
4. If the order of a transitive group is pm, p being a prime number,
the subgroup composed of all its substitutions which omit a given letter
omits pmo letters, where io >0.
5. All the substitutions of highest degree (nz) in any transitive group
generate a transitive group of degree n, which either coincides with the
original group or differs from it merely with regard to substitutions of
degree n-1.
Suggestion. Use the theorem that the average number of letters
in all the substitutions of a transitive group of degree n is n-1, while in
an intransitive group this number is smaller.
6. The total number of the substitutions which are conjugate to a given
substitution of a group must generate either the entire group or an
invariant subgroup. This theorem remains true if the word conjugate
is replaced by the word similar.
7. The number of the substitutions of degree p3 and of order p in the
symmetric group of degree n is prime to p whenever pa is the highest power
of p which does not exceed n. Cf. Annals of Mathematics, vol. 16 (1915),
p. 169.
15. Primitive and Imprimitive Groups. When the subgroup G1, composed of all the substitutions of a transitive
group G which omit a given letter, is of degree n-a, n being
the degree of G, we have seen that the letters of G may be
divided into n/a sets, each set involving a distinct letters, such
that these sets are transformed as units by all the substitutions
of G. Such a substitution group is said to be imlprimitive
whenever (> 1, and the sets of a letters are called its systems
of imprimilivity. These systems of imprimitivity are transformed under G according to a transitive group which has a