326 
ON THE THEORY OF GROUPS. 
[690 
where, for shortness, ab, ace, &c., are written instead of (ah), (ace), &c., to denote the 
cyclical substitutions a into b, b into a; and a into c, c into e, e into a, &c.; and 
it is at once seen that by the same substitutions the lines may be derived from any 
other line. 
It will be noticed that in the foregoing substitution-group each substitution is 
regular, that is, composed of cyclical substitutions each of the same number of letters; 
and it is easy to see that this property is a general one; each substitution of the 
substitution-group must be regular. 
By what precedes, the group of any order composed of the functional symbols is 
replaced by a substitution-group upon a set of letters the number of which is equal 
to the order of the group, and wherein all the substitutions are regular. 
The general theory being thus explained, I endeavour to form a substitution- 
group with the twelve letters abcdefghijkl; and I assume that there is one substitution, 
such as abc.clef.ghi.jkl, and another substitution, such as agj. bfi. cek. dhl. Observe 
that, if the twelve letters are to be thus arranged in two different ways as a set 
of four triads, without repetition of any duad, all the ways in which this can be 
done are essentially similar, and there is no loss of generality in taking the two sets 
of triads to be those just written down. But the substitution to be formed with either 
set of triads will be different according as any triad thereof, for instance agj, is written 
in this form or in the reversed form ajg. There are thus in all sixteen substitutions 
which can be formed with the first set of triads, and sixteen substitutions which can 
be formed with the second set of triads; and the relation of a triad of the first set 
to a triad of the second set is by no means independent of the selection of the 
triads out of the two sets respectively. To show this, take the two substitutions quite 
at random; suppose they are those written down above, say 
a = abc. def. ghi. jkl, ¡3 — agj .bfi. cek. dhl; 
and perform these in succession on the primitive arrangement il = abcdefghijkl. The 
operation stands thus: 
/Sail = fegkihlbjcda, 
ail = bcaefdhigklj, 
il = abcdefghijkl, 
whence 
/3a, = afhbeijcgl. dk, 
is not a regular substitution; and, by what precedes, a, /3 cannot belong to a group. 
But take the substitutions to be 
a. (as before) = abc. def. ghi. jkl, /3 = ajg. bif. cek. dhl, 
/Sail = iejkbhlfacdg, 
ail = bcaefdhigklj, 
il = abcdefghijkl, 
then we have