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

82

ABSTRACT GROUPS

[CH. III

as a non-regular transitive substitution group unless it contains
a non-invariant subgroup H which involves no invariant subgroup of G besides the identity.
This condition is sufficient as well as necessary in order
that G can be represented as a non-regular transitive substitution group. In fact, if the elements of G are arranged in a
rectangle, where those of such a subgroup H appear as the first
row, as follows:
SI, S2,... Sh
S1t2 2t2,..., Sht2
sltn, S2tn...' Shtn
the lines are permuted as units if all the elements are multiplied
on the right by any element of G, since these lines are the cosets of G as regards H.
Hence each element of G may be denoted by the substitution according to which it permutes these co-sets when it is used
as a multiplier in the given manner. No two elements of G
could permute these co-sets according to the same substitution,
since H is non-invariant under G and does not involve any
invariant subgroup, besides the identity, of G. This proves
the following theorem: A necessary and sufficient condition
that an abstract group G of order g can be represented as a transitive
substitution group of degree n is that G contains a non-invariant
subgroup of order gin which does not include any invariant subgroup of G besides the identity.
In the given method of representing G as a transitive substitution group of degree n it is clear that H corresponds to the
subgroup composed of all the substitutions which omit a given
letter of this transitive group. All the subgroups of G which
correspond to H in one of the possible automorphisms of G
give rise to the same transitive substitution group, but no other
subgroup can have this property. Hence G gives rise to as
many different transitive groups of degree n as it has sets of
subgroups of order gin such that each set includes all those
subgroups which correspond in some one of the possible auto