Let be a module. This means that is an abelian group
equipped with a left action of , i.e., a group homomorphism
, where denotes the group of bijections
that preserve the group structure on . Alternatively, is a
module over the ring
in the usual sense of module. For
example,
with the trivial action is a module over any group ,
as is
for any positive integer . Another example
is
, which acts via multiplication on
.
For each integer there is an abelian group
called the *th cohomology group of acting on *. The
general definition is somewhat complicated, but the definition for
is fairly concrete.
For example, the *0th cohomology group*

for all

is the subgroup of elements of that are fixed by every element
of .
The *first cohomology group*

is the group of -cocycles modulo -coboundaries, where

such that

and if we let
denote the set-theoretic map
,
then

There are also explicit, and increasingly complicated, definitions of
for each in terms of certain maps
modulo a subgroup, but we will not need this.
For example, if has the trivial action, then
, since
for any . Also,
. If
, then since is finite there are no nonzero
homomorphisms
, so
.

If is any abelian group, then

is a -module. We call a module constructed in this way
*co-induced*.
The following theorem gives three properties of group cohomology,
which uniquely determine group cohomology.

We will not prove this theorem. For proofs see
[Cp86, Atiyah-Wall] and
[Ser79, Ch. 7]. The properties of the theorem
uniquely determine group cohomology, so one should in theory be able
to use them to deduce anything that can be deduced about cohomology
groups. Indeed, in practice one frequently proves results about
higher cohomology groups by writing down appropriate exact
sequences, using explicit knowledge of , and chasing diagrams.

*Remark 11.2.2*
Alternatively, we could view the defining properties of the theorem
as the definition of group cohomology, and could state a theorem
that asserts that group cohomology exists.

*Remark 11.2.3*
For those familiar with commutative and homological algebra, we have

where

is the trivial

-module.

*Remark 11.2.4*
One can interpret

as the group of equivalence classes of
extensions of

by

, where an extension is an exact sequence

such that the induced conjugation action
of

on

is the given action of

on

.
(Note that

acts by conjugation, as

is a normal
subgroup since it is the kernel of a homomorphism.)

**Subsections**
William Stein
2012-09-24