Second cohomology group for nontrivial group action of Z4 on Z4
This article gives information about the second cohomology group for a specified nontrivial action of the group cyclic group:Z4 on cyclic group:Z4. The elements of this classify the group extensions with cyclic group:Z4 as an abelian normal subgroup and cyclic group:Z4 the corresponding quotient group.
The value of this cohomology group is cyclic group:Z2.
Get more specific information about cyclic group:Z4 |Get more specific information about cyclic group:Z4|View other constructions whose value is cyclic group:Z2
Description of the group
Consider the homomorphism that sends to the inverse map of . Accordingly, is the identity map and is also the inverse map of .
We are interested in the second cohomology group for the action of on , i.e., the group:
The cohomology group is isomorphic to cyclic group:Z2.
We consider here the two cohomology classes, with representative cocycles for each. For simplicity, we choose a representative cocycle that is a normalized 2-cocycle, i.e., if either of the inputs is the identity element of , the output is the identity element of .
|Cohomology class type||Number of cohomology classes||Representative cocycle||Corresponding group extension||GAP ID second part (order is 16)||Base characteristic in whole group?|
|trivial||1||everywhere||nontrivial semidirect product of Z4 and Z4||3||No|
|nontrivial||1||if , where we choose||M16||6||Yes|