FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 273 
Let us denote the exponents of g^, g^ by 2^\ [When \ is odd these are 
2'"-!, 2'""^ 2^, and when \ is even, 2'^~\ 2'"“b 2^.] 
First suppose that any two of the generators belong to the same group. 
Say gi and g^ both belong to the same group ; 5^ and 5^ may be equal or unequal. 
If 5 ^ z= So, Lemma (ii.) shows that the exponent oig-^g^ divides 2^^~\ therefore 
= 1 [n^od (1 + f)^], 
gf' ~ V/' ~= 1 [mocl (1 + ifl 
where 
2 ^ 1 -'^ ^ 0 (mod 2'^'), 
2 'Si-i ^ 0 (mod 2'’^), 
therefore g^, g^, cannot belong to the same group. 
If 52 = ^3 + cr, then p'/ has exponent and Lemma (ii.) shows that 
i/r"Vr"""=l[mod(l+f)^], 
where 
2'5i-i = 0 [mod (1 + ^)^], 
therefore g^ and y, cannot belong to the same group. 
Hence the three generators must belong to three difiPerent groups. 
Next suppose that the three generators belong to three groups, such as a, yS, y. 
Then 
= a^y= l [mod (l + f)^J, 
where 
2 'Si -1 ^ 0 (mod 2''^), 
2 'Sa-i ^ 0 (mod 2''^), 
2.S3-1 ^ Q (niod 2^’), 
and therefore the three generators must not belong to three such groups. 
Finally, we can see that if the three generators are taken from three different 
groups, excluding the seven sets of three groups, then 
gi^' 92^^- 9-6^" = 1 [mod (1 + f)"] 
is not possible unless 
ii = 0 (mod 2'’i), 
= 0 (mod 2'’"), 
jig = 0 (mod 2^“). 
Lemma (i.) shows that g9\ g^^, g^^ belong to the same groups as do g^, g^, g^- 
Suppose tliat these are a, yS, and S. 
MDCCCXCIII.—A. 
2 N 
