On the Relatively Abelian Corpora. 29 



Ave obtain the three numbers 



1 + ç^- = o, ij^^-"-=4. + 3,0, 1 + ,-,- • = 4 - 3/s 



wliich belong (mod. (1 + 2/0") to the exponents 



respectively. Thus €^^ = ?>, whence we conclude that the above 

 three numbers form a system of l)ases of SI. The exponents to 

 Avhich they belong are the invariants of 3( in this case. 



As for 33, the invariant being '1, we may take —1 as its base. 



(iv) The natural prime 2 is itself a prime number in /•(/>), and 



d = h /=2, Ä = l. 



Thus we have another case where d is divisible l^y 7>— 1. 

 If n — \. then there exist primitive roots of 2, e.g. p. 

 If ??=2, then ?-=:2; the invariants of 3( being 2, 2. Putting 

 - = -'2, ^^=\, l^=p, 



we obtain a system of bases of 5f : 



-1, 1-2^. 



If 7?>2, then r = 3. Putting 



- = -2, ?! = 1 , I: = [', c, = -fr, 

 we obtain the three numl)ers 



l+^^- = -l, 1 + Ij:= 1-2o, 1 + ç~=1-V, 



That these numbers form a system of bases of 3( can be shewn as 

 in (iii). The invariants of 31 in this case are 



2, 2"-\ 2"--. 



Since the last one (1— 4/''') of the above bases can 1)e decomposed 

 into 1—2/' and 1 + 2;', we can represent all the elements of the 

 group also l)y means of 



-1, l-2^o, l + 2,o. 



But these three do not form a s^^stem of bases. 



As for ^. the invariant being 3, we may take p as its base. 



