This article gives a basic definition in the following area: group cohomology
View other basic definitions in group cohomology |View terms related to group cohomology |View facts related to group cohomology
Definition
Let
be a group acting on an abelian group
, via an action
. Equivalently,
is a module over the (possibly non-commutative) unital group ring
of
over the ring of integers.
Definition in cohomology terms
The second cohomology group
(also denoted
) is an abelian group defined in the following equivalent ways.
When
is understood from context, the subscript
may be omitted in the notation for the cohomology group, as well as the notation for the groups of 2-cocycles and 2-coboundaries.
No. |
Shorthand |
Detailed description of , the second cohomology group
|
1 |
Explicit, using the bar resolution |
, is defined as the quotient where is the group of 2-cocycles for the action and is the group of 2-coboundaries.
|
1' |
Explicit, using the normalized bar resolution |
Same as definition (1), but we use normalized cocycles and normalized coboundaries instead of arbitrary cocycles and coboundaries.
|
2 |
Complex based on arbitrary resolution |
Let be a projective resolution for as a -module with the trivial action. Let be the complex . The cohomology group is defined as the second cohomology group for this complex.
|
3 |
Complex based on arbitrary injective resolution (works if category of -modules has enough injectives!) |
Let be an injective resolution for as a -module with the specified action . Let be the complex where has the structure of a trivial action -module. The cohomology group is defined as the second cohomology group for this complex.
|
4 |
As an functor |
where is a trivial -module and has the module structure specified by .
|
5 |
As a right derived functor |
, i.e., it is the second right derived functor of the invariants functor for (denoted ) evaluated at . The invariants functor sends a -module to its submodule of elements fixed by all elements of .
|
All these definitions have natural analogues for the
cohomology group
for all
. For more, see cohomology group.
Definition in terms of group extensions
There is an alternative definition of
that is specific to 2 and has no easy analogue for other
. This is in terms of group extensions.
can also be identified with the set of congruence classes of group extensions with normal subgroup isomorphic to
and quotient group isomorphic to
where the induced action of the quotient is the specified action
. By a group extension, we mean a group
having
as a normal subgroup and
as a quotient group. Two extensions
and
are congruent if there is an isomorphism of
to
which is identity on
and induces the identity map on
as a quotient.
Equivalence of the definitions
Further information: Equivalence of definitions of second cohomology group
Particular cases
A very special case where a lot of additional things of interest happen is that where the action is trivial. See second cohomology group for trivial group action. In particular, in the case of a trivial action, the second cohomology group coincides with the second cohomology group of the classifying space of
with coefficients in
.
Functoriality and automorphisms
Covariance in the second group
Suppose
is a group and
are abelian groups. Suppose
,
, and
are group homomorphisms such that
for all
.
In other words,
is a homomorphism from the
-module
with action
to the
-module
with action
.
Then, we get an induced homomorphism between the second cohomology groups:
This association is functorial, i.e., it gives a (covariant) functor from the category of
-modules (i.e., abelian groups with
acting on them) to the category of abelian groups.
Contravariance in the first group
Further information: restriction functor on cohomology, inflation functor on cohomology
Suppose
are groups and
is an abelian group. Suppose
,
, and
are homomorphisms such that
, i.e., the
-action and
-action on
are compatible. Then, we get an induced homomorphism between the second cohomology groups:
Note that the direction of this homomorphism is reverse to the direction of the original homomorphism. The association gives a contravariant functor. The functor in general is termed the restriction functor.
Automorphism group actions
- Due to the covariance in the second argument, there is a natural action on
of the group
, i.e., the subgroup of the automorphism group of
comprising those automorphisms that commute with the action of
.
- Due to the contravariance in the first argument, there is a natural action on
of the subgroup of
that send every coset of the subgroup
to itself (or equivalently, induce the identity map on
. Here,
is a normal subgroup of
defined as the kernel of
.
Examples
Here, we use the notation with
a group acting on an abelian group
via a group action
.
Extreme examples
- If
is a trivial group, then the second cohomology group
is also a trivial group.
- If
is a trivial group, then the second cohomology group
is also a trivial group.
Other examples
Acting group |
Group acted upon |
Action |
Second cohomology group |
Groups obtained as extensions |
More information
|
cyclic group:Z2 |
cyclic group:Z2 |
trivial action |
cyclic group:Z2 |
Klein four-group and cyclic group:Z4 |
second cohomology group for trivial group action of Z2 on Z2
|
cyclic group:Z2 |
cyclic group:Z4 |
trivial action |
cyclic group:Z2 |
direct product of Z4 and Z2 and cyclic group:Z8 |
second cohomology group for trivial group action of Z2 on Z4
|
cyclic group:Z2 |
cyclic group:Z4 |
non-identity element acts by inverse map |
cyclic group:Z2 |
dihedral group:D8 and quaternion group |
second cohomology group for nontrivial group action of Z2 on Z4
|
cyclic group:Z2 |
Klein four-group |
trivial action |
Klein four-group |
elementary abelian group:E8, direct product of Z4 and Z2 (occurs in three ways) |
second cohomology group for trivial group action of Z2 on V4
|
cyclic group:Z2 |
Klein four-group |
non-identity element acts by exchanging coordinates |
trivial group |
dihedral group:D8 |
second cohomology group for nontrivial group action of Z2 on V4
|
cyclic group:Z4 |
cyclic group:Z2 |
trivial action |
cyclic group:Z2 |
direct product of Z4 and Z2 and cyclic group:Z8 |
second cohomology group for trivial group action of Z4 on Z2
|
Klein four-group |
cyclic group:Z2 |
trivial action |
elementary abelian group:E8 |
elementary abelian group:E8, direct product of Z4 and Z2 (3 times), dihedral group:D8 (3 times), quaternion group |
second cohomology group for trivial group action of V4 on Z2
|
cyclic group:Z2 |
cyclic group:Z8 |
trivial action |
cyclic group:Z2 |
direct product of Z8 and Z2 and cyclic group:Z16 |
second cohomology group for trivial group action of Z2 on Z8
|
cyclic group:Z2 |
cyclic group:Z8 |
non-identity element acts by inverse map |
cyclic group:Z2 |
dihedral group:D16 and generalized quaternion group:Q16 |
second cohomology group for inverse map action of Z2 on Z8
|
cyclic group:Z2 |
cyclic group:Z8 |
non-identity element acts by cube map |
trivial group |
semidihedral group:SD16 |
?
|
cyclic group:Z2 |
cyclic group:Z8 |
non-identity element acts by the fifth power map |
trivial group |
M16 |
?
|
cyclic group:Z4 |
cyclic group:Z4 |
trivial action |
cyclic group:Z4 |
direct product of Z4 and Z4, cyclic group:Z16 (occurs in two ways), direct product of Z8 and Z2 |
second cohomology group for trivial group action of Z4 on Z4
|
Klein four-group |
cyclic group:Z4 |
trivial action |
elementary abelian group:E8 |
direct product of Z4 and V4, direct product of Z8 and Z2, central product of D8 and Z4, M16 |
second cohomology group for trivial group action of V4 on Z4
|
Klein four-group |
cyclic group:Z4 |
one coordinate acts by inverse map, other coordinate acts trivially |
? |
direct product of D8 and Z2, direct product of Q8 and Z2, central product of D8 and Z4, dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16 |
|
Klein four-group |
Klein four-group |
trivial action |
elementary abelian group:E64 |
elementary abelian group:E16, direct product of Z4 and Z4, direct product of Z4 and V4, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, direct product of D8 and Z2, direct product of Q8 and Z2 |
second cohomology group for trivial group action of V4 on V4
|
cyclic group:Z8 |
cyclic group:Z2 |
trivial action |
cyclic group:Z2 |
direct product of Z8 and Z2 and cyclic group:Z16 |
second cohomology group for trivial group action of Z8 on Z2
|
direct product of Z4 and Z2 |
cyclic group:Z2 |
trivial action |
elementary abelian group:E8 |
direct product of Z4 and V4, direct product of Z8 and Z2, direct product of Z4 and Z4, SmallGroup(16,3), M16, nontrivial semidirect product of Z4 and Z4 |
second cohomology group for trivial group action of direct product of Z4 and Z2 on Z2
|
dihedral group:D8 |
cyclic group:Z2 |
trivial action |
elementary abelian group:E8 |
direct product of D8 and Z2, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, dihedral group:D16, generalized quaternion group:Q16, semidihedral group:SD16 |
second cohomology group for trivial group action of D8 on Z2
|
quaternion group |
cyclic group:Z2 |
trivial action |
Klein four-group |
direct product of Q8 and Z2, nontrivial semidirect product of Z4 and Z4 |
second cohomology group for trivial group action of Q8 on Z2
|
elementary abelian group:E8 |
cyclic group:Z2 |
trivial action |
elementary abelian group:E64 |
elementary abelian group:E16, direct product of Z4 and V4, direct product of Q8 and Z2, central product of D8 and Z4, direct product of D8 and Z2 |
second cohomology group for trivial group action of E8 on Z2
|
Klein four-group |
cyclic group:Z8 |
trivial action |
elementary abelian group:E8 |
direct product of Z8 and V4, central product of D8 and Z8, direct product of Z16 and Z2, M32 |
second cohomology group for trivial group action of V4 on Z8
|