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

~ 45] GROUPS CONTAINING ONLY ABELIAN SUBGROUPS 113

abelian subgroups and hence it would be invariant under the
entire group. Consequently this primitive group of degree n
would involve exactly n-1 substitutions of degree n, while
each of its remaining substitutions besides the identity would
be of degree n-1.
The substitution groups which have these properties have
been studied extensively. Frobenius * proved that such a
group must have an invariant subgroup of order n. This important fact will be proved in ~ 139. If we assume this theorem
for the present, it results that G must contain an invariant
subgroup of index p, since the quotient group with respect to
the given subgroup of order n is abelian.
We shall now prove that the order g of G cannot be divisible
by more than two distinct prime numbers. Suppose that
g=plalp2a2... x, where pi, P2,..., px are distinct prime
numbers. Since G contains an invariant abelian subgroup of
prime index, we may suppose that it contains an invariant
subgroup H of order h, where h is given by the formula
h=plal-lp2a2... pX.
As H is the direct product of its Sylow subgroups and as every
operator of G which is not in H has an order which is divisible
by pi, it results directly that G contains only one Sylow subgroup of each of the orders p2a2,.., pXaX.
Let s be any operator, which is found in G but not in H
and whose order is of the form pif. As s transforms each of
the Sylow subgroups of H into itself, and as G is non-abelian,
s must be non-commutative with some of the operators in one
of these Sylow subgroups. This Sylow subgroup and s must
generate G, otherwise G would involve a non-abelian subgroup.
'This completes the proof of the fact that the order of a nonabelian group which contains only abelian subgroups cannot be
divisible by more than two distinct prime numbers.
Suppose that the order of G is plalp2a, a2> 0, and that G
contains an invariant subgroup of order plal- p2a2, and represent
the Sylow subgroups of orders plal and p2a2 by P1 and P2 respec* Frobenius, Berliner Sitzungsberichte, 1902, p. 455.