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

60

ABSTRACT GROUPS

[CH. III

can be obtained by combining the elements in every possible
manner. This fact results from the equation
t sattst = satSYt 8tl + = sJt2
Hence the theorem: If t transforms a group G into itself and if
t' is the lowest power of t which occurs in G, then t and G generate
a group whose order is 6 times the order of G. This theorem is
very useful in the construction of groups.
The theorem which has just been proved can be readily
extended by replacing the cyclic group generated by t by any
group H. It has been observed that all the common elements
of G and H constitute a subgroup of both of these groups,
viz., the cross-cut of G and H. By replacing the first column
of the given rectangle by elements from the different co-sets
of H as regards this cross-cut, we arrive at a more general
theorem which may be stated as follows: If all the elements
of a group H transform G into itself, then H and G generate a group
whose order is the order of G multiplied by the index under H
of the cross-cut of G and H. It is easy to verify that all the
elements of the group generated by G and H transform G into
itself. Hence G is an invariant subgroup of this group.
It was observed in ~ 10 that when 1, s2, sa,..., sy is
a subgroup of the group G it is always possible to arrange
all the elements of G in both of the following ways so that
no element is repeated:
1, s2, s3,..., Sy 1, S2 S3 ~ v.S3 Sy
t2, S212, S3t2, ~ ~ e, St2 t2h t2S2, t2S3, * *., t2Sy
S2t, S3t,..., S.yt t., tXS2, tXs3,..., tXSy
In these arrangements the elements of the first column do not
necessarily constitute a group. By interchanging rows and
columns it becomes evident that such an arrangement is
possible when the elements of the first row do not form a group.
The question arises whether all the elements of G can be
arranged in the given manner even when neither the first row
nor the first column is a subgroup of G. That such an arrange