Second cohomology group for trivial group action of Z4 on Z4
Description of the group
where and .
The cohomology group is isomorphic to cyclic group:Z4.
Note that since cyclic over central implies abelian, the fact that the top group is cyclic forces all the corresponding group extensions to be abelian. In particular, all the 2-cocycles are symmetric 2-cocycles.
|Cohomology class type||Number of cohomology classes||Representative 2-cocycle||Corresponding group extension||Second part of GAP ID (order is 16)|
|trivial||1||everywhere||direct product of Z4 and Z4||2|
|symmetric, nontrivial, order two||1||takes the value when the inputs' integer values add up to less than , otherwise||direct product of Z8 and Z2||5|
|symmetric, nontrivial order four||2||takes the value when the inputs' integer values add up to less than , otherwise||cyclic group:Z16||1|