# Group cohomology of dihedral group:D8

View group cohomology of particular groups | View other specific information about dihedral group:D8

## 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 first few homology groups are given below:

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

### Over an abelian group

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

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

## 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

The first few cohomology groups are given below:

$q[itex] !! [itex]0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$
$H^q$ $\mathbb{Z}$ 0 $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z}$  ?  ?  ?  ?

### Over an abelian group

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

$q[itex] !! [itex]0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$
$H^q$ $M$ $\operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2) || [itex]M/2M \oplus M/2M \oplus \operatorname{Ann}_M(2)$  ?  ?  ?  ?  ?

## 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 cyclic group:Z2.

This has implications for projective representation theory of dihedral group:D8.

The three possible Schur covering groups for dihedral group:D8 are: dihedral group:D16, semidihedral group:SD16, and generalized quaternion group:Q16. For more, see second cohomology group for trivial group action of D8 on Z2, where these correspond precisely to the stem 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 D8 and Z2, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16 second cohomology group for trivial group action of D8 on Z2
cyclic group:Z4 4 1  ?  ? second cohomology group for trivial group action of D8 on Z4