232 Isomorphisms of Groups whose Degree is less than Eight. 
the direct product o£ the symmetric groups of orders 6 and 
24 respectively. The group of order 40 is simply isomorphic 
with its I, since it is the direct product of the group of order 
2 and a complete group which involves only one subgroup 
of half its order. The group of order 48 contains a charac- 
teristic subgroup of order 8 and two invariant subgroups of 
order 6. From these subgroups it follows that the order of 
its I is 96, and this I is the direct product of this group of 
order 48 and the group of order 2. 
Since two of the groups of order 72 are direct products of 
characteristic subgroups, their I's are the direct products 
of the groups of isomorphisms of these subgroups. The 
orders of these Fs are 48 and 144 respectively. The re- 
maining group of order 72 involves three invariant tetra- 
hedral groups and contains all the operators of its I which 
transform each of these invariant subgroups into itself. As 
these subgroups are transformed according to the symmetric 
group of order 6 by I, it follows that the order of this group 
of isomorphisms is 432. With respect to the characteristic 
abelian subgroup of order 12, it is isomorphic to the direct 
product of two symmetric groups of order 6, since the 
group of cogredient isomorphisms is isomorphic with the 
symmetric group of order 6 with respect to the same 
subgroup, and does not permute the given three tetrahedral 
groups. 
The group of order 120 is the direct product of two 
characteristic subgroups, and its I is the symmetric group of 
order 120. Since the group of order 144 is the direct 
product of two characteristic subgroups which are also com- 
plete groups, it is simply isomorphic with its I. The simple 
group of order 168 is known to have the group of degree 8 
and order 336 for its I, while the group of order 240 is 
simply isomorphic with its I, according to the given theorem. 
The remaining two groups are the alternating and the sym- 
metric, and hence have the latter for their common I. Hence 
the groups of isomorphisms of the 29 groups, which may be 
represented on seven but on no smaller number of letters, have 
the following orders : 4, 6, 12, 42, 40, 144, 24, 48, 16, 96, 
432, 120, 336, 240, 5040. 
University of Illinois. 
