# Group of IIP 2-cocycles for trivial group action is naturally identified with group of homomorphisms from a particular abelian group

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## Statement

Suppose $G$ is a group. Then, there exists an abelian group $K$ such that, for any abelian group $A$, the group $Z^2_{IIP}(G,A)$ of IIP 2-cocycles $\! f:G \times G \to A$ for the trivial group actioncan be identified with the group of homomorphisms $\operatorname{Hom}(K,A)$ under pointwise addition.

## Examples

Group $G$ Group $K$ that acts as source of homomorphisms
cyclic group:Z4 group of integers
Klein four-group cyclic group:Z4
cyclic group:Z8 $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$
direct product of Z4 and Z2 $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z} \times \mathbb{Z}$
elementary abelian group:E8 direct product of Z4 and Z4 and Z4 and Z2 ($\! \mathbb{Z}_2 \times \mathbb{Z}_4^3$)
cyclic group:Z16 $\! \mathbb{Z}^7$
direct product of Z4 and Z4 $\! \mathbb{Z}^6 \times \mathbb{Z}_2 \times \mathbb{Z}_4$
direct product of Z8 and Z2 $\! \mathbb{Z}_2^2 \times \mathbb{Z}^6$
direct product of Z4 and V4 $\! \mathbb{Z}_2^5 \times \mathbb{Z}_4 \times \mathbb{Z}^4$
elementary abelian group:E16 $\! \mathbb{Z}_2^5 \times \mathbb{Z}_4^6$
elementary abelian group:E32 $\! \mathbb{Z}_2^{16} \times \mathbb{Z}_4^{10}$