CHAPTER XVI. 
SOME APPLICATIONS OF THE THEORY OF GROUPS OF 
LINEAR SUBSTITUTIONS AND OF GROUP-CHARACTERISTICS. 
239. We shall now apply some of the methods and results 
of the preceding three chapters to obtain a series of special 
theorems some of which give properties of a group independent 
of its mode of representation, while others are directly con 
cerned with permutation-groups. 
The theorem, due to Prof. Frobenius, that a transitive 
permutation-group whose operations except E permute all or 
all but one of the symbols, contains a self-conjugate sub 
group whose order is equal to its degree, and the theorem that 
every group whose order contains only two distinct primes is 
soluble, are good examples of the power of this method. Before 
the development of the theory of group-characteristics these 
theorems, though special cases of them had been established, 
presented difficulties which had not been overcome. It cannot 
be doubted that further important results await the investigator 
in this line. 
240. It has been seen in § 225 that i i® an algebraic 
integer. If m is the order of the operations of the ¿th set, x% is 
the sum of Xi wth roots of unity; and unless these are all the 
same, mod. Xilx? zero or a rea l positive quantity less than 
unity. This is immediately obvious when the roots of unity 
are represented graphically. 
If S is any operation of the ¿th set, and if /r is a number less 
than and prime to m, then belongs to a conjugate set of h 
operations, whose characteristic is obtained from x% on replacing 
b. 21