294 
MATHEMATICS: G. A. MILLER 
We shall now prove that when p is any odd prime number it is possible to 
construct a non-abelian group whose order is of the form p m and which admits 
a (1 + &^)-holomorphism, k being an abitrary positive integer. Suppose that 
k is divisible by pP~ x but not by pP. Hence 1 + kp = 1 + hpP, where h is 
prime to p. Let t be an operator of order p which transforms an operator s 
of order pP +l into its (1 + hjP) th power. It will be proved that t transforms 
each operator of the non-abelian group G of order p& +2 which is generated by s 
and / into its (1 + hpP) th power. 
This proof is an almost direct consequence of the two dependent equations 2 
{sty = /, (st)^ = y 
In fact, from these equations it results that t transforms sPrfPx and s a i into 
the same powers. If we form the direct product of the group G just con- 
structed and any abelian group of type (1,1,1, . . . ) we clearly obtain an- 
other non-abelian group which is such that / transforms each of its operators 
into a (1 + -holomorphism. The group G can therefore be used to con- 
struct an infinite system of groups each of which admits such a holomorphism. 
To prove the theorem under consideration it is desirable to note that it is 
possible to construct a non-abelian group whose order is of the form 2 m and 
which admits a (1 + 2 7 ) -holomorphism whenever y > 2. In fact, if s is an 
operator of order 2 T+1 and if / is an operator of order 2 which transforms s 
into its (1 + 2 7 Y h power the non-abelian group of order 2 7+2 which is gener- 
ated by s and / will clearly satisfy the required condition. Moreover, each one 
of the infinite system of groups obtained by forming the direct product of the 
group just constructed and an abelian group of order 2 l and of type (1,1,1, ... ) 
must likewise satisfy this condition. 
It is now easy to establish the general theorem noted in the second para- 
graph. To construct a non-abelian group which admits an a-holomorphism, 
a>3, it is only necessary to consider the factors of a — 1. When a — 1 is 
of the form 2 w any one of the groups described in the preceding paragraph satis- 
fies the condition in question. When a — 1 is not of this form let p be any 
one of its odd prime divisors and suppose that a — 1 is divisible by jr but not 
by Hence a is of the form 1+ hpP considered above and it has been 
proved that whenever a>3 it is possible to construct an infinite system of groups 
such that each group of this system admits an a-holomorphism. 
1 J. W. Young, New York; Trans. Amer. Math. Soc, 3, 1902, (186). 
2 Miller, Blichfeldt and Dickson, Theory and Applications of Finite Groups, Wiley, New 
York, 1916, p. 108. 
