# Second cohomology group for nontrivial group action of Z4 on Z4

From Groupprops

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

Let be cyclic group:Z4 with generator and be cyclic group:Z4 with generator . Note that the automorphism group is isomorphic to cyclic group:Z2, with its non-identity element being the inverse map.

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.

## Elements

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 |