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148 7. Galois Cohomology and Isomorphism Classification of Elliptic Curves

(2.6) Proposition. Let G be a group, and let E be a G-group with a G-subgroup A.
Applying the fixed element functor to the exact sequence 1 → A → E → E/A → 1,
we obtain an exact sequence of pointed sets
G G G
1 → A → E → (E/A) .
Proof. Since A G → E G is the restriction of a monomorphism A → E,it is a
monomorphism. Next,we calculate

ker E G → (E/A) G = ker(E → E/A) ∩ E G

G
= im(A → E) ∩ E G = im A → E G .
This proves the proposition.
The above simple proposition brings up the question of the surjectivity of E G →
G
(E/A) . If we try to establish surjectivity,we would consider a coset xA fixed by G,
s
that is, (xA) = xA for all s ∈ G. Then we must determine whether or not xA can
s
s
be represented by x ∈ E with x = x. All we know is that x ∈ xA for all s which
s
means that x = xa for some a ∈ A. We can view any coset X in E/A as a right A-
set with the multiplication E × E → E on E inducing the action X × A → X which
s
s s
is G-equivariant,that is,on which (xa) = x a. Observe that X has the property

that for two points x, x ∈ X there exists a unique a ∈ A with xa = x . This
property says that X is a right A-principal homogeneous G-set,and these objects are
considered in the next section. Returning to the question of the exact sequence (2.6),
we will study the question of when the right A-set X has a G-invariant point in this
context.
§3. Principal Homogeneous G G G-Sets and the First Cohomology Set
1
H H H (G G G, A A A)
In this section G denotes a group. Following our analysis at the end of the previous
section on cosets relative to a G-subgroup A,we make the following definition.

(3.1) Definition. Let A be a G-group. A principal homogeneous G-set X over A is
a right A-set X with a left G-set structure such that:
s
s s
1. The right A action on X defined X×A → X is G-equivariant,i.e., (xa) = x a
for all s ∈ G, x ∈ X,and a ∈ A.

2. For any two points x, x ∈ X there exists a ∈ A with x a = x ,and further a ∈ A

is unique with respect to this property.
If A is a G-subgroup of a G-group E,then any coset X ∈ E/A which is G-
invariant,that is, s(X) = X for all s ∈ G,is an example of a principal homogeneous
G-set over A as we observed at the end of the previous section. In particular, A acting
on itself by group multiplication is a principal homogeneous G-set,and in this case,
there is at least one fixed element,the identity.