2-coboundary for trivial group action

Suppose $G$ is a group and $A$ is an abelian group. A 2-coboundary for trivial group action for $G$ on $A$ is a 2-coboundary for the trivial group action of $G$ on $A$.

Explicitly, it is a function $f:G\times G\to A$ satisfying the condition that there exists a function $\alpha :G\to A$ such that:

$f(g,h)=\alpha (g)+\alpha (h)-\alpha (gh)\ \forall \ g,h\in G$

$f$ may be called the coboundary of $\alpha$. Conceptually, $f$ measures the extent to which the function $\alpha :G\to A$ deviates from being a homomorphism. If $\alpha$ were a homomorphism, $f$ would be the zero function.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness	Capture of difference	Intermediate notions
constant 2-cocycle for trivial group action	a constant function $c_{a}:G\times G\to A$ that sends all inputs to a fixed element $a\in A$	We can set $\alpha$ as the constant function $G\to A$ sending everything to $a$		The quotient is isomorphic to the group of normalized 2-coboundaries with canonical splitting	|FULL LIST, MORE INFO
normalized 2-coboundary for trivial group action	$f(1,1)=0$; equivalently, it is obtained from a function $\alpha$ satisfying $\alpha (1)=0$			The quotient is isomorphic to the base group $A$ with canonical splitting ($A$ being constant functions)	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness	Capture of difference	Intermediate notions
2-cocycle for trivial group action	$f(g,hk)+f(h,k)=f(g,h)+f(gh,k)$			The quotient is the second cohomology group for trivial group action $H^{2}(G,A)$	|FULL LIST, MORE INFO