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

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PRIMITIVE AND IMPRIMITIVE GROUPS

39

(1, m) correspondence with G. If m> 1, G must therefore contain an invariant subgroup of order m which transforms each
of these systems of imprimitivity into itself. If m= 1, each
substitution of G, besides the identity, must transform at least
one of these systems of imprimitivity into another.
While every transitive group whose G1 omits more than one
letter is necessarily imprimitive, it does not follow that the G1
of an imprimitive group must omit more than one letter. We
proceed to prove that a necessary and suflicient condition that G
is imprimitive is that G1 is contained in a larger subgroup of G.
In other words, a necessary and sufficient condition that G is
imprimitive is that G1 is a non-maximal subgroup of G. It
should be observed that this theorem connects the theory of
imprimitivity with the theory of abstract groups. Every
transitive substitution group which is not imprimitive is said
to be primitive.
The given theorem is contained in a more general theorem
which may be stated as follows: A necessary and sugicient
condition that a complete set of conjugate substitutions or subgroups of G is transformed under G according to an imprimitive
substitution group is that the largest subgroup of G which transforms into itself one of these substitutions or subgroups is contained
in a larger subgroup of G. That this condition is sufficient
results from the fact that if this largest subgroup K, which
transforms into itself a substitution or subgroup L, is contained
in a larger subgroup H, then the number of the conjugates of
L under H is equal to the quotient obtained by dividing the
order of H by the order of K, and H involves all the substitutions of G which transform these conjugates among themselves.
Every substitution of G which is not in H must therefore
transform the set of conjugates of L under H into an entirely
new set of conjugates, and therefore this set of conjugates is
transformed as a unit. Hence all the conjugates of L can be
divided into sets such that they are all transformed as units
and such that no two sets have a common substitution or subgroup. On the other hand, if the complete set of conjugates