Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

36

36 ~~~SULBSTITUIJTON G-ROUIP (i. I

I-CH..11

The permutation of the substitutions of each ol( they e rows
as regards the first row represents a substitution, and ree may
assume, without loss of generality, that the 1vubstr tutiorls represented by the first square, when the various Trows a re associated
successively with the first row, are the soosti-i-o ~no of GC Any
one of these rows, say the one invol aing Sa, rep1r ser(is a subs-titution obtained by multiplying all th'e suhstiitr-itioas ofi G ona the
right by sa,; while any row of CG', say tr one_~rjvohnngjj So,
represents a substitution obtained b- multiplying al I- substitutions of C- on the left by s03. Since we get thec sam e result
when we multiply all the substitutions of C first on -oh Wright by
s5a, and then on the left by sp as when w,,e multualy tem first
on the left by s, and then on -the right by s,, as Ca co-,ns. quen-ce of
the associative law, it follows that eacla Subs~tcuin of G is
commutative with every substitut-ion of C'" and -u;,ce versa.
Moreover, G' includes all the substitutions on thlesec letters,
which are commutative with every suibstit-u-Ci of G, sinoe
every such substitution must involve alr the le"IJIr fs or G and
the totality of these substitutions forms a group.
As G and C' are different ways of represenLiny the group
G they must be simply isomorphic. It remains t-o proven thiat
they are conjugate. If we establish a simrple oorhs
between G and C' in such a way that- all the substiuiLuons begin
with the same letter, the second letters in all. tL-h Subs-I-rtutions
of these groups represent the substitution by mnearts or which
C may be transformed so that the first two letrcr5 ~n each
one of its substitutions are the same as -ho t 5 o letters
of the corresponding substitution iit G' O S61ce thirs r -ansfoLrmation will lead to simply isomorphic. grourps, aod si-ce two
simply isomorphic regular groups have rthc -woperty dltr thcorresponding substitutions are identical whenrecer the flirst
two letters of all these substitutions are the same0, we have
proved Jordan's theorem; viz., with every~ reg tar grot p o
order n there is associated another regular YrouP of orhdei nisuecz
that each of these groups is composed of the tota' (Hi uonber oj" substitutions on these nt letters which are coww~rleUhl rgeve substit~ution of the other group.