TITLE: Motivation for the relations defining $H^1(G,A)$ for non-commutative cohomology
QUESTION [7 upvotes]: First let me review the definition of first non-commutative cohomology. Let $G$ be a group and $A$ a left $G$-group, i.e. for any $\sigma, \tau\in G$ and $a, b\in A$, one has $\sigma(\tau(a))=(\sigma\tau)(a), \sigma(ab)=\sigma(a)\sigma(b)$. 
A 1-cocycle from $G$ to $A$ is a function $f: G\to A$ such that $f(\sigma\tau)=f(\sigma)\sigma(f(\tau))$ for any $\sigma, \tau\in G$. Two 1-cocycles $f, g$ are equivalent if there exists $c\in A$ such that $f(\sigma)=c^{-1}g(\sigma)\sigma(c)$ for all $\sigma\in G$, this is easily verified to be an equivalence relation. Then the first non-commutative cohomology $H^1(A,G)$ is defined to be the set of 1-cocycle modulo equivalence relation. This is a pointed set with base point given by the trivial cocycle.
My question is: both relations $f(\sigma\tau)=f(\sigma)\sigma(f(\tau))$ and $f(\sigma)=c^{-1}g(\sigma)\sigma(c)$ seem to be unnatural. Does there exists a good motivation for these relations?

REPLY [4 votes]: Here is one application that might provide some motivation. Given a finite group $G,$ one might consider a representations of $G.$ These are finite dimensional $\mathbb{F}$-vector spaces $V$ on which $G$ acts as linear transformations. But what is a linear transformation? 
You may be thinking it's a matrix. But you'd be wrong. 
A technical point that you may have glossed over when first learning linear algebra is that matrices are representations of linear transformations with respect to a chosen basis. Said in another way, given a basis $\beta$ of $V$ one obtains a homomorphism from $[\cdot]_{\beta}: End(V) \rightarrow M_n(\mathbb{F})$ from the ring of linear transformations of $V$ to $n$ by $n$ matrices over $F.$ And given a different basis $\beta'$ one may obtain a different homomorphism $[\cdot]_{\beta'}.$ Luckily, these homomorphisms are related by conjugation by the change of basis matrix $M_{\beta, \beta'}.$ Now one might like to think of a representation of $G$ as a homomorphism from $G$ to $GL_n(\mathbb{F}),$ but in order to associate such a map to a representation one must first choose a basis and in general there is no best choice. To avoid such a choice one instead associates to the representation the set of all homomorphisms from $G$ to $GL_n(\mathbb{F})$ obtained upon choosing a basis for $V.$ This set of homomorphisms is exactly a $GL_n(\mathbb{F})$ conjugacy orbit in $Hom(G,GL_n(\mathbb{F}))$ i.e. an element of  $H^1(G,GL_n(\mathbb{F}))$  where $G$ acts trivially on $Gl_n(\mathbb{F}).$ On the other hand, to any element of $c \in H^1(G,GL_n(\mathbb{F}))$ we may associate the representation given by any lift of $c$ to $C^1(G,GL_n(\mathbb{F})).$ One checks that these maps are inverses and thus we see there is a bijection 
$$H^1(G,GL_n(\mathbb{F})) \cong \{\text{n-dimensional Linear Representations of G over } \mathbb{F}\}.$$ 
One can broaden the category of representations of $G$ by considering semilinear representations. These are finite dimensional vector spaces $V$ together with an action of $G$ on $V$ and $\mathbb{F},$ such that 

$g(cv)=g(c)g(v)$ for all $c \in \mathbb{F}$ and $v \in V$
$g(v_1 + v_2)=g(v_1) + g(v_2)$ for all $v_1,v_2 \in V.$
$G$ acts on $\mathbb{F}$ as field automorphisms 

One can then tell a similar story as in the case of linear representations. Given a semilinear representation $V$ of $G$ one obtains for every basis $\beta$ of $V$ a map $f_{\beta}:G \rightarrow GL_n(\mathbb{F})$ which sends $g$ to $[g]_{\beta}.$ One checks that such a map satisfies $f_{\beta}(g_1g_2) = f_{\beta}(g_1)g_1(f_\beta(g_2))$ and for a second basis $\beta'$ of $V$ the two maps $f_{\beta}$ and $f_{\beta'}$ are related by $M_{\beta,\beta'}^{-1}f_{\beta}(g)g(M_{\beta,\beta'}) = f_{\beta'}.$ Associating to $V$ the set of all such maps we obtain an element of $H^1(G,GL_n(\mathbb{F}))$ where the action of $G$ on $GL_n(\mathbb{F})$ is induced from the action of $G$ on $\mathbb{F}.$ Calling the action of $G$ on $\mathbb{F}$ $\Sigma.$ It follows by an identical argument as above  
$$H^1(G,GL_n(\mathbb{F})) \cong \{\text{n-dimensional Semi-Linear Representations of G over } \mathbb{F} \text{ where G acts on } \mathbb{F} \text{ by } \Sigma\}.$$