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

146

GROUPS DEFINED ABSTRACTLY

[CH. VI

group for a quotient group but not for a subgroup. If a set
of generating operators satisfy certain conditions, the largest
group which they may generate has all the other groups which
they may generate for quotient groups. This follows directly
from the theory of the quotient group. When m=n=2, G
is clearly abelian and is of order 4 or 8. In the former case it
is cyclic and in the latter it is of type (2, 1). When m=3
and n= 2, G is the dicyclic group of order 12.
When n =l, the category of groups under consideration
clearly coincides with the dihedral groups. When n=2 and m
is odd it coincides with all the dicyclic groups whose orders
are not divisible by 8, and when n = 2 and m is even, the groups
of order mn coincide with the totality of the dicyclic groups,
while those of order 2mn may be obtained by establishing an
(m, 2) isomorphism between either the dihedral group of order
2m or the dicyclic group of order 2m and the cyclic group of
order 4. Hence the dihedral groups and the dicyclic groups
may both be regarded as special cases of groups generated
by two operators having a common square. Since S1S2=
SlS2-ls22, it results that the order of the product of the two
operators sl and s2 is either the least common multiple between
m and n, or it is exactly half this least common multiple. That
is, the order of the product of two operators is not restricted
by the fact that they have a common square, and the order
of the group which they generate is always a divisor of the
double of the square of the order of this product.
If we let tl=sl and t2=sS12-1, it results that the group
(tl, t2) is identical with (si, s2). That is, every group that can
be generated by two operators having a common square can also
be generated by two operators such that the one transforms the
other into its inverse, and vice versa. Hence we have two abstract
definitions for this category of groups. The latter definition
is more convenient than the former for the purpose of obtaining directly the abstract properties of these groups, but in the
abstract theory these groups frequently present themselves
under the former definition, and hence it is very important
to know that the two given definitions apply to the same cate