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

156

GROUPS DEFINED ALBSTRACTLY

[CH. VI

S12=S23, (s1S2)4 = 1, is the direct product of the group of order
5 and the group or order 96 obtained by extending the nontwelve group of order 24 by means of an operator of order 8
which transforms it according to an outer isomorphism of
order 2.
It is evident that the group of order 48 obtained by establishing a (2, 12) isomorphisr between the cyclic group of order
4 and the octahedron group is generated by two operators of
orders 4 and 6 respectively which satisfy the given equations.
Moreover, the direct product of this group of order 48 and the
group of order 5, and the direct product of the octahedral
group and this group of order 5 contain two generating operators which satisfy the conditions under consideration. Hence
we have arrived at the theorem:
There are exactly six non-abelian groups which can be generated
by two operators which fufil the equations 12= S23, (s1S2)4=1.
Three of these are of orders 24, 48, and 96, respectively, while
the other three are the direct products of these respective groups and
the group of order 5.
A second generalization of the octahedron group is given
by the equations
S13 = S24, (S1S2)2 = 1.
Since the two operators SlS2, S2S1 are of order 2, they generate
a dihedral group. To determine an upper limit of the order of
this group we observe that
(S2S12S2)3 = (s22s22)3 S9 = 121
As si21 is invariant under G, its order is a divisor of 42, and an
upper limit of the order of this dihedral group is evidently 12,
while the order of G is a divisor of 336. When Si isof order 6
the order of G is 48. Moreover, it is easy to see that the group
of order 48, which may be obtained by extending the nontwelve group of order 24 by an operator of order 2 which-transforms it according to an outer isomorphism, is generated by two
operators of orders 6 and 8 respectively, which satisfy the given
conditions. This group may be represented transitively on
eight letters, and it involves 12 operators of each of the orders