Second cohomology group for trivial group action
Contents
Definition
Context for definition
Let be a group and
be an abelian group.
In terms of more general definition
The second cohomology group for trivial group action is defined as the second cohomology group for the trivial group action of on
. This group is denoted
.
Note that is also used for the more general notion of second cohomology group with an accompanying action. The notation is interpreted in terms of the trivial group action only if that is explicitly stated or is otherwise clear from context.
Definition in terms of explicit 2-cocycles and 2-coboundaries
The second cohomology group, denoted , is defined as the quotient
where
is the group of 2-cocycles for the trivial group action and
is the group of 2-coboundaries for the trivial group action.
Definition in terms of group extensions
can also be identified with the set of congruence classes of central extensions of
by
, i.e., group extensions where the normal subgroup
is a central subgroup and the quotient group is
.
Definition in terms of cohomology of classifying space
Suppose is the classifying space of
. The second cohomology group for trivial group action
is defined as the second cohomology group
where the latter is in the sense of the cohomology of a topological space (for instance, singular or cellular cohomology).
Group actions on the second cohomology group
- Automorphism group of base group acts on second cohomology group for trivial group action: Note that there is a corresponding statement for a nontrivial group action, but in that more general case, we can only make the subgroup
act.
- Automorphism group of acting group acts on second cohomology group for trivial group action
Subgroups of interest
Some subgroups of interest are:
- IIP subgroup of second cohomology group for trivial group action
- cyclicity-preserving subgroup of second cohomology group for trivial group action
Facts
Examples
- Second cohomology group for trivial group action of finite cyclic group on finite cyclic group
- Second cohomology group for trivial group action commutes with direct product in second coordinate: There is a natural isomorphism: