Group cohomology of elementary abelian group of prime-square order
Suppose is a prime number. We are interested in the elementary abelian group of prime-square order .
Contents
Homology groups
Over the integers
The first few homology groups are given below:
Failed to parse (syntax error): (\mathbb(Z}/p\mathbb{Z})^2 = E_{p^2} | ||||||
rank of as an elementary abelian -group | -- | 2 | 1 | 3 | 2 | 4 |
Over an abelian group
The homology groups with coefficients in an abelian group are given as follows:
Here, is the quotient of by and .
These homology groups can be computed in terms of the homology groups over integers using the universal coefficients theorem for group homology.
Cohomology groups for trivial group action
Over the integers
The cohomology groups with coefficients in the integers are given as below:
The first few cohomology groups are given below:
0 | ||||||
rank of as an elementary abelian -group | -- | 0 | 2 | 1 | 3 | 2 |
Over an abelian group
The cohomology groups with coefficients in an abelian group are given as follows:
Here, is the quotient of by and .
These can be deduced from the homology groups with coefficients in the integers using the dual universal coefficients theorem for group cohomology.