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}