GROUPS OF ORDER P^ WHICH CONTAIN CYCLIC 

 SUB-GROUPS OF ORDER P-^ <'^ 



By L. I. Neikirk 



The groups of order p"', which contain self-conjugate cyclic 

 sub-groups of orders p""' and p"""^ respectively, have been deter- 

 mined by Burneide, Theory of Groups of a Finite Order, pp. 75-81, 

 and the number of groups of order p", which contain cyclic non self- 

 conjugate sub-groups of order p""^ has been given by Miller, Trans- 

 actions of the American Mathematical Society, Vol. Ill, No. 4 and 

 Vol. II, No. 3. Prof. Miller has used a method which partially de- 

 pends on a special form of representation of these groups, i. e., as 

 substitution groups. 



The method of treatment used in this paper is entirely abstract 

 in character, and in virtue of its nature, it is possible in each case to 

 give explicitly the generational equations of these groups. They are 

 divided into two classes, and it will be shown that these classes cor- 

 respond to the two partitions (m-2, 2) and (m-2, 1, 1). 



Assume an abstract group G of order p™. G contains operators 

 of order p""* and no operator of greater order exists in G. Let P 

 denote one of the operators of G of order p"'~^ The p^ power of 

 every operator in G is contained in the cyclic sub-group |P|, other- 

 wise the order of G would be greater than p". 



The division into classes is effected by the following assump- 

 tions : 



1" There is in G at least one operator Q, whose p power is not 

 contained in | P | . 



2" The p power of every operator in G is contained in | P } . 



[p is assumed an odd prime and m is taken greater than 4 the 



(') Thesis presented to the Graduate Faculty of the University of Colorado for the degree 

 of Master of Science. 



