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

34

SUBSTITUTION GROUPS

[CH. 1I

stituent group K, then each of these k substitutions is found
in the same number of the substitutions of G, and hence k is a
divisor of g, where g is the order of G. In the simple isomorphism method, k=g. All the substitutions of G which involve
only the identity from K constitute an invariant subgroup
H of G, and K is said to be a quotient group of G as regards H.
This quotient group is commonly represented by the following
symbol: *
G/ = oK
For instance, consider the following intransitive group G:
I, cde, ced, ab cd, ab de, ab ce.
One of the transitive constituent groups is 1, ab-K, and the
invariant subgroup of G,. which involves only the identity
of this constituent group, is as follows:
H-=1, cde, ced.
The quotient group G/tH- K may also be regarded as a group
in which the three substitutions of H are regarded as one substitution, while the remaining three substitutions constitute
the other substitution. The other transitive constituent of
G is simply isomorphic with G, and the given correspondence
is sometimes represented by the following symbol:
1
cde 1
ced
cd
de ab
ce
As an instance of a more general correspondence, we may
consider the group of order 18 composed of all the positive
substitutions in the direct product of the symmetric group of
degree 3 represented on two distinct sets of letters. This
group is represented as follows:
* This symbol was used by C. Jordan, Bulletin de la Societe Malhenalique de
France, vol. 1 (1872), p. 46. Thie syTmbol is often credited to To6lder, who used
it in 1889. Cf. HI. Weber, Kleines Lehrblmch dcr 'I Igebra,: 1912, p. 1 92.