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

182

SOLVABLE GROUPS

[CH. VIII

Since G involves operators of orders pi"a, the number of its
operators whose orders divide g/pi is less than g. This number
is known to be a multiple of g/pi (~ 25) and hence it can be
written in the form kg/pl, where k is an integer. The number
of operators of G whose orders are divisible by pi"a is therefore equal to
g-kg/pi =l(pi -1),
since this number is also a multiple of the number of the different possible generators of a cyclic subgroup of order pial. The
first member of the given equation is divisible by g/pi, and,
as each of the prime factors of this divisor exceeds pi-1, it
results that I is also divisible by g/pi. Hence k = 1, and
l=g/pl.
If al> 1, it can be proved, in exactly the same way, that the
number of the operators of G whose orders are divisible by
pia-l but not by pial is
g/pi-kig/pi2 =l (pi-1).
Hence kl=l, and ll=g/pi2. By continuing this process it
results that the number of operators of G whose orders divide
paf.. p /x, 3 <X, is exactly equal to this number. In particular, G contains only one Sylow subgroup of order pxx,
and the corresponding quotient group contains only one subgroup of order px-_lx-1, etc. Hence G contains a cyclic quotient group of order pi"l, and the invariant subgroup of G
which corresponds to the identity in this quotient group is
such that each of its Sylow subgroups, with the possible
exception of those of order px"x, is cyclic. This completes a
proof of the following theorem:
If all the Sylow subgroups whose orders are not divisible by the
highest prime which divides the order of the group are cyclic, the
group is solvable.
In particular, every group whose order is not divisible by
the square of a prime number is solvable. Hence there is only
one such group when none of these primes diminished by unity
is divisible by another.