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

~ 33] ABSTRACT GROUP REPRESENTED BY SUBSTITUTIONS 81

hence it establishes the fact that it is impossible to construct
a group G such that its order g is divisible by p", but not by
pa+l and such that the number of its operators whose orders
divide pa lies between p" and p+l.
EXERCISES
1. If the number of the operators of a group G, whose orders divide
an arbitrary divisor d of the order of G, is exactly d, then G must be
cyclic.
2. There are no groups with the property that every cyclic subgroup
besides the identity is transformed into itself by only its own operators.
Cf. Dyck, Mathematische Annalen, vol. 22 (1883), p: 101.
3. Let x1 and n represent two arbitrary numbers, and denote by si
and S2 the operations of subtracting n from x1 and dividing x2 by n respectively. If n is replaced successively by all the numbers resulting from
these operations, then sl and s2 will, in general, generate the symmetric
group of order 6 when 12 =x2. When xl2=2x2, and when x12= 3x2, these
operations will generate the octic group and the dihedral group of order
12 respectively.*
33. Representation of an Abstract Group as a Transitive
Substitution Group. In ~ 27 it was observed that every group
of finite order can be represented in one and only one way as a
regular substitution group. It is often very useful to represent
a given abstract group as a transitive substitution group on the
smallest possible number of letters. Many of the abstract properties of a group can often be most readily determined if the
group is written in this form. Hence we proceed to consider
the general question of representing an abstract group G as a
non-regular transitive substitution group of degree n.
In ~ 12 it was observed that when G is thus represented, it
involves n conjugate subgroups G1, G2,..., Gn each of which
is composed of all the substitutions of G which omit a given
letter. If the subgroup of G1, composed of all its substitutions
which omit a given letter, is of degree n-a, these n conjugate
subgroups are identical in sets such that each set involves a
of them. As G is non-regular, there must be at least two such
sets, and hence we see directly that G cannot be represented
* These groups, together with the four group, have been called groups of subtraction and division, Quarterly Journal of Mathematics, vol. 37 (1906), p. 80._