Does anyone here know how to compute
H^3(G,Z)
where G is the klein 4 group?
To compute the third integral cohomology group H^3(G, Z) of the Klein 4-group G, we can use group cohomology theory. This involves determining the set of group cochains and the corresponding coboundary maps.
First, let's break down the process step-by-step:
Step 1: Determine the group cochains
Group cochains are functions that assign an integer to each n-tuple of elements from the group G. In this case, G is the Klein 4-group, which has four elements: the identity element (e), and three other elements (a, b, c) such that each element squared is equal to the identity. Therefore, the group cochains are functions that assign an integer to each 4-tuple of elements (e, a, b, c).
Step 2: Define the coboundary map
The coboundary map is a function that takes a group cochain and produces a new cochain with an extra index. In this case, we are interested in the third cohomology group, so we need to define the coboundary map that maps from the space of second cochains to the space of third cochains. This map is given by the group commutator operation, [g, h] = g⁻¹h⁻¹gh, where g, h are elements of the group G.
Step 3: Compute the coboundary set
To compute the coboundary set, we apply the coboundary map to the group cochains determined in Step 1. This gives us a set of 4-tuples, each representing a third cochain.
Step 4: Identify the cohomology group H^3(G, Z)
Finally, the third integral cohomology group H^3(G, Z) is the set of equivalence classes of third cochains, where two cochains are considered equivalent if their difference is a coboundary (i.e., one can be obtained from the other by applying the coboundary map).
By following these steps, you should be able to compute H^3(G, Z) for the Klein 4-group G.