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

168

ISOMORPHISMS

[CH. VII

would be invariant under a group having the symmetric group
of degree 3 as a transitive constituent. As the group of isomorphisms of the group of order p is cyclic, this would imply
that the subgroup, composed of all the substitutions in the
primitive group of degree n and index less than n+1, which
transform a subgroup of order p into itself, would involve
substitutions of the form ab or abc. As this is impossible, it has
been proved that the alternating group of degree n, n>7,
cannot involve a transitive subgroup of degree n and of index
less than n+1. Hence the symmetric group of degree n, n>7,
cannot contain a subgroup of index less than n+l and greater
than 2 except the symmetric group of degree n-1.
From what precedes, it results that the only subgroups
of index n in the symmetric and the alternating groups are those
of degree n-1, whenever n>7. This theorem is known to
be true also as regards the groups of degree 7. From this
fact and from the theorem in ~ 67, it follows directly that the
group of isomorphisms of the alternating and of the symmetric
group of degree n, n> 6, is the symmetric group of this degree.
EXERCISES
1. Prove that in an automorphism of the symmetric group of degree
6 substitutions of the form abc may correspond to those of the form abc def,
and that all the operators of order 3 in this symmetric group are conjugate under its holomorph.
2. Prove that the symmetric groups of degrees 4 and 5 are complete
groups, and that the alternating groups of these degrees have the corresponding symmetric groups for their groups of isomorphisms.
3. Give an instance of a group which involves an invariant subgroup
whose group of isomorphisms is larger than that of the entire group.
66. Several Useful Theorems Relating to the Groups of Isomorphisms.* Every abelian group can be extended so that we
obtain a group of twice the order of the original group, by
means of operators of order 2 which transform each operator
of this abelian group into its inverse. These groups may be
regarded as a direct generalization of the dihedral groups, and
may therefore be called generalized dihedral groups as regards
* Cf. Philosophical Magazine, vol. 231 (1908), p. 223.