Take the 2-minute tour ×  When reviewing my notes and Serre's
book "Galois Cohomology" Chapter 5 dealing with non-abelian
group cohomology, I realized that I don't fully understand the
concept of biprincipal spaces such as a-(A,B)-space (defined in
page 49).  I think that what Serre ment is that is a G-set P
with two G-groups A and B such that:  $\forall p,q \in P :
\exists ! a \in A : q=a*p$  $\forall p,q \in P : \exists !b \in
B : q=p*b$  $\forall p \in P , a \in A , b \in B:
(a*p)*b=a*p*b=a*(p*b)$ (A and B action commutes)  Then he talks
about composition of such spaces: if P is a (A,A')-space and Q
is a (A',A'')-space then  $P \circ Q = P \times ^{A'}Q $  which
he claims has a (A',A'')-space structure. If so, for what is
this composition is good for? Q is already a (A',A'')-space.
Serre defines $ P \times ^AF$ as follows: let A be a G-group,
let P be a principal homogeneous space on which A acts from the
right and F a G-set on which A acts from the left. Then it is a
quotient space of $P \times F$ induce