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Group cohomology of direct product of Z4 and Z2

Homology groups for trivial group action

Cohomology groups for trivial group action

FACTS TO CHECK AGAINST (cohomology group for trivial group action):
First cohomology group: first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms
Second cohomology group: formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization
In general: dual universal coefficients theorem for group cohomology relating cohomology with arbitrary coefficientsto homology with coefficients in the integers. |Cohomology group for trivial group action commutes with direct product in second coordinate | Kunneth formula for group cohomology

Over the integers

H^q(\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}; \mathbb{Z}) = \left \lbrace \begin{array}{rl} (\mathbb{Z}/2\mathbb{Z})^{(q-1)/2}, & \qquad q = 1,3,5,\dots \\ (\mathbb{Z}/4\mathbb{Z}) \oplus (\mathbb{Z}/2\mathbb{Z})^{q/2}, & \qquad q = 2,4,6,\dots \\ \mathbb{Z}, & \qquad q = 0\end{array}\right.

The first few cohomology groups are given below (these are the same as the first few homology groups, shifted over by one):

q 0 1 2 3 4 5
H_q \mathbb{Z} 0 \mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \mathbb{Z}/2\mathbb{Z} \mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}

Over an abelian group

Below are the cohomology groups with coefficients in an abelian group M:

H^q(\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z};M) = \left \lbrace \begin{array}{rl} (M/2M)^{(q-1)/2} \oplus \operatorname{Ann}_M(4) \oplus \operatorname{Ann}_M(2)^{(q+1)/2}, & \qquad q = 1,3,5,\dots \\ (M/4M) \oplus (M/2M)^{q/2} \oplus \operatorname{Ann}_M(2)^{q/2}, & \qquad q = 2,4,6,\dots \\ M, & \qquad q = 0\end{array}\right.

Here, M/2M represents the quotient of M by the subgroup 2M = \{ 2x \mid x \in M \}, M/4M represents the quotient of M by the subgroup 4M = \{ 4x \mid x \in M \}, \operatorname{Ann}_M(2) = \{ x \in M \mid 2x = 0 \} and \operatorname{Ann}_M(4) = \{x \in M \mid 4x = 0 \}.

Second cohomology groups and extensions