60 



Art. 5.— T. Takenouclii ; 



All the cyclic subgroups of this Abclian group are of the form 



(Är''S'f-' ST' s',f"r, c=o,i,2, ,p-i, 



Avhere a^,(u, ,«i, ,«•,, are some fixed integers for each sub- 

 group. Each of these a's can take p values: 0,1/2, ,2^ — ^. 



Hence there are (j^ — 1)''"^ special cyclic subgroups, in which all of 

 these a' s are different from zero. If Ave denote by ^ a divisor 

 of A belonging to one of these special cyclic subgroups, then B 

 must possess the following properties : 



(i) A>B^ C[C', C',, 



(ii) A is of thepth relative degree with respect to B, 



(iii) none of the original cychc corpora C'l, C., , C,, is 



contained in B completely. 



It can. easily be seen that these three properties are characteristic 

 of B. 



Let us call B a derived corpvs of C\, C., , C,,. There are 



in all (i)-l)''"^ different derived corpora. In particular, when 79=2, 

 the derived corpus is uniquely determinate. 



II. Let there be systems of cyclic corpora 



C„C,, , A 



A, A. 



E,,E,, 



<ph, qd'2, 



)of degrees^ '•'^' ^■'''-' 



r'l, v% 



J \ 



respectively, where p, cp , r, 11, v, are all primes. Suppose 



that these corpora are independent of one another, and that 



21, q, , r are all different from one another, but not necessarily 



all different from u, v, Composing these corpora all together, 



Ave obtain an Abelian corpus, which Ave shall call A. 

 Now, let 



C-i, 0-2, , ^-h, A, , Jl^i, Jli-j, 



