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

~ 56] TWO GENERATORS HAVING A COMMON SQUARE 145

have a common square and that the order of sls2-~ is m.
The order of the group generated by these two operators is
always a divisor of 2mn and a multiple of mn, since s12 and
slS2-1, are of orders n and m respectively and the cyclic group
generated by these operators cannot have more than two
common operators.
It results from the preceding paragraph that G is completely
defined by the three conditions
512 =S22, (S1S2-1)m 1, s2=1
whenever either m or n is odd. In fact, when si, S2 fulfil these
conditions and either m or n is an odd number, G is the group
of order 2mn obtained by establishing an (m, i) isomorphism
between the dihedral group of order 2m and the cyclic group
of order 2n. If m and n are both even, G may again be constructed by establishing such an (m, n) isomorphism when
the order of G is 2mn. When this order is only ma, G may
be constructed by establishing a simple isomorphism between
n cyclic groups of degree and order m, and then extending this
group by means of an operator of order 2n which transforms
into its inverse each operator of this cyclic subgroup, permutes
its systems of intransitivity, and has its nth power in this subgroup. That is, the two operators si, s2 may be so selected as to
fulfil the three conditions
S12=22, (S1S2-')m =, S12=1
and to generate either of two groups when m and n are both given
even numbers. When at least one of them is a given odd number,
the group generated by si, s2 is completely determined by the three
given relations.
To illustrate this theorem we begin with the case when
m =4 and n =2. The group of order mn in this case is clearly
the quaternion group, while the group of order 2mn may be
constructed by establishing a (4, 2) isomorphism between the
octic group and the cyclic group of order 4 so as to obtain the
group of order 16 involving 12 operators of order 4 which have
only two distinct squares. This group has the quaternion