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

~ 43]

CONFORMAL GROUPS

107

43. Abelian Groups which are Conformal with Non-abelian
Groups. Two distinct groups are said to be conformal when
they contain the same number of operators of each order.*
We proceed to determine all the abelian groups which are conformal with non-abelian groups. The complete solution of
the converse of this problem, viz., the determination of all the
non-abelian groups which are conformal with abelian ones, is
much more difficult, since a large number of distinct nonabelian groups may be conformal with the same abelian group,
while no more than one abelian group can be conformal with
one non-abelian group. In fact, it was observed in ~ 37 that
two distinct abelian groups cannot be conformal.
It is evident that there is only one group of order 2m which
does not include any operator of order 4, viz., the abelian group
of type (1, 1, 1,... ). Moreover, there is only one cyclic
group of order 2m, and when m<4 no two groups of order 2m
are conformal. We proceed to prove that every abelian group
G of order 2m which does not satisfy one of these conditions is
conformal with at least one non-abelian group.
Let H be the subgroup of G which is generated by the square
of one of its independent generators s of lowest order, together
with all the other independent generators of G. The order of
H is 2m-1. Since m>3 there is an operator t of order 2 which
has the following properties: It transforms H into itself,
it is commutative with half of the operators of H (including
all those which are not of highest order), and it transforms the
rest into themselves multiplied by an operator of order 2 which
is not the square of a non-invariant operator of H; i.e., t does
not transform an operator of order 4 contained in H into its
inverse. The non-abelian group generated by H and t is conformal with G whenever s2= 1.
When the order of s exceeds two, the group generated
by t and H (written as a regular substitution group) may be
made simply isomorphic with itself by writing it on two distinct sets of letters. If in this intransitive group t is replaced
by the continued product of t, the substitution of order two
* Quarterly Journal of Mathematics, vol. 28 (1896), p. 270.