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

158

GROUPS DEFINED ABSTRACTLY

[CH. VI

of orders 36 and 72 respectively which satisfy the given condition and generate the group of order 432 in question. Hence
it is easy to deduce the following theorem:
If two non-commutative operators satisfy the conditions si2 = 24,
(s1s2)3=1, they may generate the dihedral group of order 6, the
octahedral group, the group of order 48 obtained by extending the
non-twelve group of order 24 by means of an operator of order 4
which has its square in this group and transforms it according to
an outer isomorphism of order 2, the direct products of these respective groups and a cyclic group of order 3 or 9. Hence there are
exactly nine non-abelian groups which may be generated by two
such operators.
EXERCISES
1. There are exactly six non-abelian groups whose two generators
Si, s2 satisfy the equations s12=s23, (sIs2)= 1. They are the icosahedron
group, a group of order 120, and the direct products of these respective
groups and the cyclic groups of orders 5 and 25.
2. If two commutative operators satisfy the equations S12=S25,
(slS2)3= 1, they generate a cyclic group whose order is 3, 7, or 21; if they
satisfy the equations 13= s25, (siS2)2= 1, they generate a cyclic group whose
order is 2, 4, 8, or 16; if they satisfy the equations s12=s23, (ss2)5=1,
they generate a cyclic group whose order is either 5 or 25.
3. There are exactly six non-abelian groups whose generators satisfy
the equations s13=s25, (SIS2)2= 1. They are a group of order 1920 and the
direct products of the icosahedral group and the cyclic group of order 2",
a=0, 1, 2, 3, 4.
4. If two commutative operators satisfy the equation s2= 24,(2)3 = 1,
they generate a cyclic group whose order is 2 3, 6, 9, or 18; if they
satisfy the equations 12= s23, (S3S2)4= 1, they generate a cyclic group whose
order is 2, 4, 5, 10, or 20; if they satisfy the equations s13=s24, (sIS2)2=1,
they generate a cyclic group whose order is 2, 7, or 14.