of the Groups whose Degree is less than Eight. 227 
it follows that I is simply isomorphic with a substitution 
group of degree n and that the group of cogredient isomor- 
phisms is simply isomorphic with G. This proves the theorem 
in question. 
If a substitution group G of degree n contains a subgroup 
of the same degree which leads to a representation which is 
conjugate to G, then G admits outer isomorphisms. In 
particular, if a simple group appears only once among the 
total number of substitution groups of degree n, and if it 
involves a subgroup which is both of degree and of index n, 
then it admits outer isomorphisms. For instance, the simple 
group of order 168 presents itself only once among the 
substitution groups of degree 7 and contains a subgroup of 
degree 7 and order 24. It must therefore admit outer iso- 
morphisms, as is also known from other considerations. As 
it contains only one set of 7 subgroup of order 24 and 
degree 7, and as it is not invariant under a larger group of 
degree 7, its I can be represented as an imprimitive group of 
degree 14 which involves two systems of imprimitivity and 
is of order 336. Such considerations apply to a large number 
of substitution groups and are frequently useful to obtain I. 
The group of isomorphisms of every finite group is finite. 
In fact, it is always easy to find an upper limit of the order 
of I for any known finite group, as this order cannot exceed 
the number of different ways in which a set of generating 
operators may be selected, and hence it can certainly never 
exceed (#—1) !, g being the order of the group. It is easy 
to see that there are only three groups for which I has this 
maximal order, viz. the four-group and the groups of order 2 
and 3. Assuming that ! = 1, the identity might also be 
classed among these groups. In most cases it is easy to find 
much lower upper limits for the order of I. 
§ 2. The Groups whose Degree is less than Six. 
There is only one group of degree 2, and its group of 
isomorphisms (I) is the identity. One of the two groups of 
degree three is the cyclic group of order 3 and hence has the 
group of order 2 for its 1, while the other is symmetric and 
therefore is simply isomorphic with its I. Two of the groups 
of degree 4 are cyclic. As their orders are 2 and 4, their Fs 
are the identity and the group of order 2 respectively. Two 
others are simply isomorphic with the four-group, and hence 
have the symmetric group of order 6 for their I. A third 
set of two groups of degree 4 is composed of the alternating 
and the symmetric groups* These have the symmetric group 
