Group of cyclicity-preserving 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 of cyclicity-preserving 2-cocycles for the trivial group action \! f:G \times G \to A can be identified with the group of homomorphisms \operatorname{Hom}(K,A) under pointwise addition.

Examples

Extreme examples

If G is a cyclic group or more generally a locally cyclic group, the corresponding group K is a trivial group.

Examples of 2-groups

When G is a 2-group, K is also a 2-group.

Group G Order Prime-base logarithm of order Group K that acts as source of homomorphisms Order Prime-base logarithm of order
Klein four-group 4 2 cyclic group:Z4 4 2
direct product of Z4 and Z2 8 3 direct product of Z4 and Z2 (\! \mathbb{Z}_4 \times \mathbb{Z}_2) 8 3
dihedral group:D8 8 3 direct product of Z8 and V4 (\! \mathbb{Z}_8 \times \mathbb{Z}_2 \times \mathbb{Z}_2) 32 5
quaternion group 8 3 cyclic group:Z4 4 2
elementary abelian group:E8 8 3 direct product of Z4 and Z4 and Z4 and Z2 (\! \mathbb{Z}_2 \times \mathbb{Z}_4^3) 128 7
direct product of Z4 and Z4 16 4 direct product of E16 and Z8 (\! \mathbb{Z}_2^4 \times \mathbb{Z}_8) 128 7
SmallGroup(16,3) 16 4 \! \mathbb{Z}_2^5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 1024 10
nontrivial semidirect product of Z4 and Z4 16 4 direct product of E16 and Z8 (\! \mathbb{Z}_2^4 \times \mathbb{Z}_8) 128 7
direct product of Z8 and Z2 16 4 direct product of Z4 and V4 (\! \mathbb{Z}_2^2 \times \mathbb{Z}_4) 16 4
M16 16 4 direct product of Z4 and V4 (\! \mathbb{Z}_2^2 \times \mathbb{Z}_4) 16 4
dihedral group:D16 16 4 \mathbb{Z}_2^6 \times \mathbb{Z}_{16} 1024 10
semidihedral group:SD16 16 4 direct product of E16 and Z8 (\! \mathbb{Z}_2^4 \times \mathbb{Z}_8) 128 7
generalized quaternion group:Q16 16 4 direct product of Z8 and V4 (\! \mathbb{Z}_2^2 \times \mathbb{Z}_8) 32 5
direct product of Z4 and V4 16 4 \!  \mathbb{Z}_2^4 \times \mathbb{Z}_4^3 1024 10
direct product of D8 and Z2 16 4 \! \mathbb{Z}_2^6 \times \mathbb{Z}_4^2 \times \mathbb{Z}_8 8192 13
direct product of Q8 and Z2 16 4 \! \mathbb{Z}_2^2 \times \mathbb{Z}_4^3 256 8
central product of D8 and Z4 16 4 \! \mathbb{Z}_2^4 \times \mathbb{Z}_4^3 1024 10
elementary abelian group:E16 16 4 \! \mathbb{Z}_2^5 \times \mathbb{Z}_4^6 131072 17
elementary abelian group:E32 32 5 \! \mathbb{Z}_2^{16} \times \mathbb{Z}_4^{10} 68719476736 32

Examples of other p-groups

We consider the case \! p = 3.

Group G Order Prime-base logarithm of order Group K that acts as source of homomorphisms Order Prime-base logarithm of order
elementary abelian group:E9 9 2 elementary abelian group:E27 27 3
direct product of Z9 and Z3 27 3 elementary abelian group:E243 243 5