Group cohomology of quaternion group

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This article gives specific information, namely, group cohomology, about a particular group, namely: quaternion group.
View group cohomology of particular groups | View other specific information about quaternion group

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

The homology groups are given as follows:

H_q(Q_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl}\mathbb{Z}, & q = 0 \\ \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}, & q \equiv 1 \pmod 4 \\ \mathbb{Z}/8\mathbb{Z}, & q \equiv 3 \pmod 4 \\ 0, & q \ne 0, q \mbox{ even} \\\end{array}\right.

The group is a finite group with periodic cohomology, in keeping with the other definition of being a group with periodic cohomology: every abelian subgroup is cyclic.

The first few homology groups are given below:

q 0 1 2 3 4 5 6 7 8
H_q \mathbb{Z} \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} 0 \mathbb{Z}/8\mathbb{Z} 0 \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} 0 \mathbb{Z}/8\mathbb{Z} 0

Over an abelian group

The first few homology groups with coefficients in an abelian group M are given below:

q 0 1 2 3 4 5 6 7
H_q M M/2M \oplus M/2M \operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2)  ?  ?  ?  ?  ?

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

q 0 1 2 3 4 5 6 7
H^q \mathbb{Z} 0 \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} 0  ?  ?  ?  ?

Over an abelian group

The first few cohomology groups with coefficients in an abelian group M are as follows:

q 0 1 2 3 4 5 6 7
H^q M \operatorname{Ann}_M(2) \oplus\operatorname{Ann}_M(2) M/2M \oplus M/2M  ?  ?  ?  ?  ?

Cohomology ring with coefficients in integers

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Second cohomology groups and extensions

Schur multiplier

The Schur multiplier, defined as second cohomology group for trivial group action H^2(G,\mathbb{C}^\ast), and also as the second homology group H_2(G;\mathbb{Z}), is the trivial group.

Schur covering groups

Since the Schur multiplier is a trivial group, the Schur covering group of the quaternion group is the quaternion group itself.

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 Klein four-group direct product of Q8 and Z2 (1 time), nontrivial semidirect product of Z4 and Z4 (3 times) second cohomology group for trivial group action of Q8 on Z2
cyclic group:Z4 4 1 Klein four-group direct product of Q8 and Z4 (1 time), nontrivial semidirect product of Z4 and Z8 (3 times) second cohomology group for trivial group action of Q8 on Z4
Klein four-group 4 2 elementary abelian group:E16 direct product of Q8 and V4 (1 time), SmallGroup(32,2), direct product of SmallGroup(16,4) and Z2 second cohomology group for trivial group action of Q8 on V4