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

~ 65] ISOMORPHISMS OF ALTERNATING GROUPS

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less than n, n>4, and its only subgroup of this index is the
alternating group of degree n-l, when n 6.
We begin with the proof of the latter part of this theorem,
since the former part can be readily deduced from the latter.
As the theorem of Cebysev, to which we have just referred,
applies only to all numbers greater than 6, and the groups of
degree seven are well known, we shall assume that n>7, and
prove that the alternating group of degree n does not contain
any subgroup whose index is less than n +1, with the exception
of its alternating subgroups of degree n-1 and of index n.
If such a subgroup existed it would be transitive on its letters,
since the order of an intransitive subgroup could clearly not
exceed 2- (n-2)!. As the order of an imprimitive subgroup is
evidently less than this number, the subgroup in question would
be primitive, and hence its order could not be divisible by the
highest power of 3 which divides n!, since a primitive group
of degree n does not involve a substitution of the form abc
unless it is the alternating group of degree n.
Since the order of the subgroup in question would not be
divisible by the highest power of 3 that divides n!, this order
would have to be divisible by the prime p, where n/2 <p n n-2.
Hence this subgroup would involve 1+kp conjugate cyclic
subgroups of order p. If two such subgroups had less than
p-1 common letters, we could transform one by an operator
of the other so as to obtain two such subgroups having a larger
number of common letters without having all letters in common.
This process could be repeated until two subgroups of order p
would be found having p-1 common letters, and hence the
primitive subgroup in question would itself involve primitive
subgroups of each of the degrees p, p+1,..., n. It would
therefore be at least four-fold transitive.
As the transitive subgroup composed of all the substitutions
involving a certain set of p letters would be invariant under
a group of degree p+3, which would involve two transitive constituents of degrees p and 3 respectively, and as this transitive
constituent of degree 3 would be the symmetric group of this
degree, it results that each of the cyclic subgroups of order p