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

~ 1171

THE PRIMITIVE SIMPLE GROUPS

243

various terms could therefore be rearranged in sets as explained
in that paragraph, which gives us an equation of the form
(7) A(l +a+a2+... +a-l)+B(1+f+2... + -l)
+C(1+y+y2,+... +r-1)+.. =0,
A, B, C,... being certain sums of roots of unity; a, 3, y,... primitive roots of the equations x= 1, =1, Xr=l,...
respectively; and p, q, r,... different prime numbers.
The coefficients A, B, C,... may be put into certain
standard forms. Thus, any root el occurring in any of
these sums will be assumed to be resolved into factors of primepower indices (~ 116, 4~): e=epeaer.., the root,p being
of index pm, e, of index q", etc. Furthermore, within A any
root ep will be assumed to be either unity or a root whose index
is divisible by p2. For, if it were of index p, say ep=at, we
could put 1 in its place, since
at(l.++a2+. +aP-'1) _-a+ra + -[ a2 +... +p= 1 +a+a2+ + -...
by means of the relation aV=l.. Likewise we assume that
any root ea within B is either equal to unity or is a root whose
index is divisible by q2; and so on.
To illustrate, let i be a root of index 4 and r a root of index 9 (namely
T73= ), and let p=2, q =3. Then the standard form for the expression
(8) ( —iW-1)(1-1-) +(r2C - 2 +i) ( l+C- + 2)
would be
(i3w +1)(1-1) +(T5- 1 +i)(1 +Co+ c2).
2~. We shall now make certain changes in the values of the
roots in the equation (7). First we put 0 for every root Ea,
Er,....whose index is divisible by the square of a prime
other than p2 (as r5 in the example above), leaving undisturbed the roots whose indices are not divisible by such a
square, as a, a2,..., 3,... The quantities A, B,..
are thereby changed into certain sums A', B',... The
equation (7) is still true, the vanishing sums l+ a+a2+..
+a'~l, etc., not having been affected.