Group cohomology of direct product of Z4 and Z2

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This article gives specific information, namely, group cohomology, about a particular group, namely: direct product of Z4 and Z2.
View group cohomology of particular groups | View other specific information about direct product of Z4 and Z2

Homology groups for trivial group action

FACTS TO CHECK AGAINST (homology group for trivial group action):
First homology group: first homology group for trivial group action equals tensor product with abelianization
Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier
General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

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}/4\mathbb{Z} \oplus (\mathbb{Z}/2\mathbb{Z})^{(q+1)/2}, & \qquad 1,3,5,\dots \\ (\mathbb{Z}/2\mathbb{Z})^{q/2}, & \qquad q = 2,4,6,\dots \\\mathbb{Z}, & \qquad q = 0\end{array}\right.

The first few homology groups are given below:

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

Over an abelian group

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

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

Second cohomology groups for trivial group action

Group acted upon Order Second part of GAP ID Second cohomology group for trivial group action Extensions Cohomology information
cyclic group:Z2 2 1 elementary abelian group:E8 direct product of Z4 and V4, direct product of Z8 and Z2, direct product of Z4 and Z4, SmallGroup(16,3), M16, nontrivial semidirect product of Z4 and Z4 second cohomology group for trivial group action of direct product of Z4 and Z2 on Z2
cyclic group:Z4 4 1 direct product of Z4 and V4 direct product of Z4 and Z4 and Z2, direct product of Z16 and Z2, direct product of Z8 and V4, direct product of Z8 and Z4, SmallGroup(32,24), M32, direct product of M16 and Z2, semidirect product of Z8 and Z4 of M-type second cohomology group for trivial group action of direct product of Z4 and Z2 on Z4