GROUPS OF TYPE I. 243 
of T for this group. Putting A = B= B’=0 and A, =B,= 
B,, =0 in equations I-IX of Art. 179 and dividing the oth- 
ers by IX we get the five following equations of the conditions 
for the group G,(Al): 
Keres (Ket) 
ko =khy ; kee! =k! key’; -— = yp  ; 
I, t=kk,; IT, III. 7 Aiestin gar ee 
TV. (iba) $5 =k (=k) ET th ey) Se (73) 
y, Bat _ Ment Aol Wat b= AY W=h)(s—2) A” 
> RTS SL SS ag ee aay UG BY’ 
ki (ky/—1) i (ky—1) Ay’ 
By! A! BB!’ © 
Making k’=k* and k,'=k;/, then k,’=k; equations I-V 
then reduce to the following equations of condition of group 
G,(Al),.2: 
eee eee Te = 2 an Js 
IV. (a) 5, = (k— 1) > +k (hi DS: (74) 
V ko2—1 — ha—1 Ay! _k®=1 _k-1 A” | k(k-1) (:-1) A” 
= eB ene = CATEMER i Bit 
ke (kit-1) _ k? (ki—1) Ay” 
By! Aj! By!’ 0 
If in these equations we make A,’= A’ = A/’ (III) i, 
pears and we get the Sule mans of the group G,(AA’),_.. If 
on the other hand we make a =a = a , then (IV) te 
pears and we have left the equations of the group G,(Il’),_ 
If both assumptions are made simultaneously, equations Ill 
and IV both disappear and we have left the equations of the 
BEOUp GAC AA oo. [keAV eA AW ands Al Al = Ao 
and B,'’= B” = B,'’, then equations LI, IV and V all disap- 
pear and we have left only k, = kk, which is the equation of 
condition of the group G,(AAA’),_,.. Hence we see that the 
group G,(Al),_, contains the following varieties of subgroups: 
G,(AA"),-., G,(W’),-2, G(AA’L),-2, and G,(AAA’),_,. We 
