ON THE THEORY OF GROUPS. 
327 
690] 
whence 
/3a = ai. be. cj. dk. fh. gl, 
a regular substitution; and, for anything that appears to the contrary, a, /3 may 
belong to a group. It is convenient to mention at once that these two substitutions 
do, in fact, give rise to a group; viz. the square diagram is 
a 
b 
c 
d ! e 
f 
9 
h 
i 
3 
k 
l 
b 
c 
a 
e 
f 
d 
h 
i 
9 
k 
l 
3 
c 
a 
b 
f 
d 
e 
i 
9 
h 
l 
3 
k 
d 
l 
h 
a 
9 
3 
e 
c 
k 
f 
i 
b 
e 
j 
i 
b 
h 
k 
f 
a 
l 
d 
9 
c 
f 
k 
9 
c 
i 
l 
d 
b 
3 
e 
h 
a 
y 
f 
k 
l 
c 
i 
3 
d 
b 
a 
e 
h 
h 
d 
l 
j 
a 
9 
k 
e 
c 
b 
f 
i 
i 
e 
j 
k 
b 
h 
l 
f 
a 
c 
d 
9 
j 
i 
e 
h 
k 
b 
a 
l 
f 
9 
c 
d 
k 
9 
f 
i 
l 
c 
b 
3 
d 
* h 
a 
e 
l 
h 
d 
9 
3 
a 
c 
k 
e 
i 
b 
f 
and the substitutions, obtained therefrom by writing successively each line over the 
top line, are 
1 =1, 
abc. 
def. 
ghi 
■jkl 
acb. 
dfe . 
gih , 
■ jlk 
a 2 , 
ad . 
bl.ch 
■eg, 
.fj.ik 
/3-a/3 2 , 
aeh. 
, bjd . 
cil , 
• j% 
/3 a-, 
afl. 
, bkh . 
cgd . 
. eij 
/3 2 a, 
1. 
bfi. 
eke . 
. dlh 
/3 2 , 
ahe. 
bdj . 
cli 
■fd h 
^a 2 /3a 2 , 
ai . 
be .cj 
.dk 
.fli. gl 
/3a, 
a 39 ■ 
bif . 
cek 
. dhl 
/3, 
ak . 
bg. cf. di 
. el. lij 
/3 2 a 2 , 
alf. 
bkh . 
edg , 
■ e .P 
/3 2 a/3 2 a.