Second cohomology group for nontrivial group action of Z2 on Z4
This article gives information about the second cohomology group for a specified nontrivial action of the group cyclic group:Z2 on cyclic group:Z4. The elements of this classify the group extensions with cyclic group:Z4 as an abelian normal subgroup and cyclic group:Z2 the corresponding quotient group.
The value of this cohomology group is cyclic group:Z2.
Get more specific information about cyclic group:Z2 |Get more specific information about cyclic group:Z4|View other constructions whose value is cyclic group:Z2
Description of the group
Let be cyclic group:Z2 and be cyclic group:Z4. Note that is isomorphic to the group cyclic group:Z2, and it comprises the identity automorphism and the automorphism sending every element of to its inverse. Consider the homomorphism that is an isomorphism: it sends the non-identity element of to the inverse map on , and the identity element of to the identity automorphism 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.
Let denote the non-identity element of and denote a generator for . 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 . Thus, to specify the cocycle , we need only specify .
|Cohomology class type||Number of cohomology classes||Representative cocycle||Corresponding group extension||GAP ID (second part, order is 8)||Base characteristic in whole group?|
|trivial||1||takes the value zero, i.e., is the identity element of .||dihedral group:D8||3||Yes|
Because each of the cohomology class types has size one, thereis no scope for permutation of these under any group actions.