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

~ 64] DOUBLY TRANSITIVE GROUPS OF ISOMORPHISMS 165

we shall exclude it in what follows and hence we shall assume
that si, S2 are non-commutative.
If si, S2 correspond to themselves in a given automorphism
of G, all the operators of the subgroup generated by si, S2 must
also correspond to themselves and this subgroup must include
more than two operators which are conjugate to si, s2 under I.
Hence we have as a first result:
If the group of isomorphisms of a group G can be represented
as a.doubly transitive group on letters corresponding to operators
of G, then the subgroup composed of all substitutions which omit
one letter of this doubly transitive group is either imprimitive or
it is a regular group of prime degree.
This theorem follows directly from the well-known theorem
that the subgroup which is composed of all the substitutions
which omit one letter of a non-regular primitive group of degree
n is always of degree n- 1. When G is abelian the given theorem
evidently remains true and the imprimitive group in question
involves systems of two letters each except when G is the fourgroup.
When the subgroup which is composed of all the substitutions of I which omit one letter is a regular primitive group,
the order of I is p(p+1), p being a prime, and I involves p+1
subgroups of order p. It must therefore involve an invariant
subgroup of order p+l which involves p conjugate operators
under I. That is, the subgroup of order p+l must be the
abelian group of order 2m and of type (1, 1, 1,... ). Hence
the following theorem:
If I is doubly transitive on letters corresponding to operators
of G and if the subgroup composed of all the substitutions which
omit one letter of I is primitive, then I is of order p(p+l1), p
being a prime, and it involves an invariant subgroup of order
p+1.
When I is a doubly transitive group on letters corresponding
to a set of conjugate operators of G, either all the operators of
this set are commutative or no two of them are commutative.
This results immediately from the fact that when I is doubly
transitive any two of its letters can be transformed into an