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

76

ABSTRACT GROUPS

[CH. I1I

EXERCISES
'1. The substitutions which represent the transformations of the symmetric group of order 6 into all its possible automorphisms constitutes a
group which is simply isomorphic with this symmetric group.
2. If a substitution of'order 2 transforms G1 into G2 it must also transform G2 into G1.
3. Any group G of order g that involves an invariant operator s of
order h can be extended by means of an operator t of order nh which is
commutative with every operator of G and satisfies the equation t-=s
so as to obtain a group of order gn.
Suggestion: Write G as a regular group on in distinct sets of letters
and establish a simple isomorphism between these groups. Let ti be
a substitution of order n which permutes the corresponding letters of this
intransitive group and is commutative with each of its substitutions. For
t we may use the product of t1 and the substitution s in one of the regular
constituents.
31. Group of Inner Isomorphisms. If all the elements of a
non-abelian group G are transformed by any one of its own
elements, the elements of G are permuted according to a certain
substitution. By transforming the elements of G successively
by all the elements of G there result a series of substitutions
which constitute a group known as the group of inner isomorphisms of G. This group is also called the group of cogredient
isomorphisms of G, and it is simply isomorphic with the central
quotient group of G (~ 28). A necessary and sufficient condition that G is simply isomorphic with its group of inner isomorphisms is that the central of G is the identity.
If G admits isomorphisms which cannot be obtained by
transforming all the elements of G by its own elements, they
are called outer, or contragredient, isomorphisms. In this case
the group of inner isomorphisms is clearly an invariant subgroup of the group of isomorphisms of G.
One of the most useful properties of the group of inner isomorphisms Ii of G is that I, contains the same number of Sylow
subgroups of every order as G does, provided we call the identity
a Sylow subgroup of order pm whenever the order of the group
Ii is not divisible by p. The truth of this fact becomes evident
if we observe that a Sylow subgroup of 11 has exactly the same
number of conjugates under 1i as the corresponding Sylow