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

~ 1221

'THE IPRIM1ITIVE SIMPLE GROUPS.

249

sponding to (5). After putting unity for every root of index
pk, the equation becomes the congruence:
{[VS2]p-[V]p}(al -a2)(a2-a3)(a3 -a'i) 0 (mod p)
which can be changed into the following:
{[VS2] -[V]p}q3 0
after multiplying by a suitable factor, since
(1-E)(1-E2)... (1-e-) =lim. x-1=q
x=l X-1
when e is a primitive qth root of unity.
Hence finally,
[VS2]p [V]p (mod p),
and the argument of ~ 118 is now valid.
COROLLARY. No primitive simple group can contain a
transformation of order 35, 150, or 21~. (If S1, representing
respectively S7, S13, or S30, has not three distinct multipliers,
Theorem 15 applies.)
122. The Sylow Subgroups. Consider now a primitive
group G of order go. A possible subgroup of order 52 or 72
would be abelian (~ 108, Cor.). By trial we find readily that
no such group can be constructed without violating Theorem
15 or the Corollary to Theorem 14. Again, if g is divisible by
35, we have a transformation of this order (~ 135, Cor. 3).
But this is impossible in a primitive simple group (~ 121, Cor.).
A subgroup of order 3ko is monomial, and contains an abelian
subgroup of order 3k-14 at least. Assume k >3; we then have
an abelian subgroup P' of order 324. When we construct
such a group, avoiding the invariant subgroup H/ resulting
from Theorem 14, we discover that it must contain a transformation of type T=(e, e, e02), where e3=w. Then if g is
divisible by 5 or 7 at the same time, we would have a transformation of order 154 or 210 (~ 135, Cor. 3), violating the
Corollary, ~ 121. In any event, we can have no abelian subgroup of order 3'-14 if k>3 (cf. Ex. 4, ~ 109).
Finally, a subgroup of order 2k is monomial and contains
an abelian subgroup of order 2k-1 at least. By trial we find
that k -1 2 (cf. Ex. 3, ~ 109).