228 Mr. G. A. Miller on the Groups of Isomorphisms 
of order 24 for their I, according to the well-known theorem 
quoted above. The seventh and last group of degree 4 is the 
octic group which is known to be its own I group. The 
orders of the groups of isomorphisms of the 7 substitution 
groups of degree 4 are therefore 1, 2, 6, 8, 24. 
The orders of the two cyclic groups of degree 5 are 5 and 
6 respectively, and hence their Ps are the cyclic groups of 
orders 4 and 2. Since the group of order 12 and degree 5 
is the direct product of the symmetric group of order 6 and 
the group of order 2 it is simply isomorphic with its I, 
according to theorem IV. This is also true of the non-cyclic 
group of order 6, as this is simply isomorphic with the 
symmetric group of this order. The metacyclic and the 
semi-metacyclic groups of orders 20 and 10 repectively are 
known to have the former for their I, while the alternating 
and the symmetric groups have the latter for their common I. 
Hence the orders of the groups of isomorphisms of the 8 sub- 
stitution groups of degree 5 are 2, 4, 6, 12, 20, 120. 
§ 3. The Groups of Degree Six, 
In the preceding section we considered all the possible 
abstract groups whose order is less than 7, and hence we 
know the groups of isomorphisms of the 9 substitution 
groups of degree 6 whose orders do not exceed their degree. 
There are ten other groups of this degree which are simply 
isomorphic with groups of lower degree, viz. three of order 8, 
two of order 12, three of order 24, and one of each of the 
orders 60 and 120. Hence only 18 of the 37 groups of 
degree 6 are distinct, as abstract groups, from those of lower 
degrees. Two of these 18 are simply isomorphic with the 
group of order 8 which contains 7 operators of order 2, and 
hence have the simple group of order 168 for their I. The 
remaining group of order 8, (abcd)cyc.(ef)), is of type (2, 1), 
and hence has the octic group for its I. The group of order 
.9 is of type (1, 1), and hence has the transitive group of 
degree 8 and order 48 which involves operators of order 8 as 
its I. Since the group of order 16 is the direct product of 
the octic group and the group of order 2, its I is known * to 
be the transitive group of degree 8 and order 64 which 
Oayley denoted by 
(ae .bf.cg. d7i)N \(abcd) 8 (efgh) s } dim. 
The group of order 18, which is the direct product of the 
* Quarterly Journal of Mathematics, vol. xxviii. (1896) p. 252. 
