First cohomology set with coefficients in a non-abelian group
The first cohomology set is defined as follows:
- Call a function to be a 1-cocycle if .
- Declare two functions to be equivalent if there exits such that . This defines an equivalence relation on functions from to .
- is defined as the quotient of the set of all 1-cocycles by the equivalence relation.
Note that when the group is an abelian group, this just becomes the first cohomology group, because:
- The 1-cocycles form a group, see 1-cocycle for a group action.
- The equivalence relation is equivalent to differing by a 1-coboundary, and the 1-coboundaries form a subgroup.
- Thus, the quotient set by the equivalence relation is the quotient group of the group of 1-cocycles by the subgroup of 1-coboundaries, which gives the first cohomology group.