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XIX. The Groups oj Isomorphisms of the Groups whose 
Degree is less than Eight. By G. A. Miller *. 
THE main objects of the present paper are to develop 
some fundamental theorems relating to the group 
of isomorphisms of a known group, especially when it is 
represented as a substitution group, and to give a complete 
list of the groups of isomorphisms of all the abstract groups 
which may be represented on seven or a smaller number of 
letters. As the groups of low degrees present themselves 
most frequently, it is believed that such a list, if it is entirely 
reliable, will render good service. Fifty-four different ab- 
stract groups can be represented on seven or a smaller number 
of letters, while the number of distinct substitution groups 
on these letters is 95. A complete list of these groups is 
found in the ' American Journal of Mathematics/ vol. xxi. 
(1899) p. 326. In this list the distinct abstract groups are 
denoted by Greek letters. 
In accord with common usage the group of isomorphisms 
will be denoted by I, while the group under consideration 
and its holomorph will generally be represented by G and K 
respectively. The following known theorems are frequently 
used : — If a group is generated by two characteristic subgroups 
which luive only the identity in common, its I is the direct 
product of the Vs of these two characteristic subgi*oups> and its 
K is the direct product of their K.'s t- In particular, the K 
of an abelian group of order p«\ p°*, p^, . . . (p 1: , p 2 , p 3 , . . . 
being different prime numbers) is the direct product of the 
X's of the subgroups of orders p°\ p" 2 , p° 3 , .... The 
symmetric group of degree n, n •=£■ 2 or 6, is simply isomorphic 
with its ,1, and the alternating group of degree n, n=£3, has the 
same group of isomorphisms as the symmetric group of the 
same degree. When n = 2 the I is identity, and when n=6 
it is a well known imprimitive group of degree 12 and 
order 1440 J. It should also be remembered that the I of 
the cyclic group of order p a , p being an odd prime, is the 
cyclic group of order p a ~ l (p — 1), and that the I of the 
cyclic group of order 2 a , »>1, is the direct product of the 
group of order 2 and the cyclic group of order 2 a ~ 2 . 
* Communicated by the Author. 
f Transactions of the American Mathematical Society, vol. i. (1900) 
p. 396. 
X Bulletin of the American Mathematical Society, vol. i. (1895) 
p. 258 ; Holder, Mathematische Annalen, vol. xlvi. (1895) p.- 345. 
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