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A crossed homomorphism from $G$ to a $G$-module $M$ is a map $f:G\to M$ satisfyingThus crossed homomorphisms are precisely one-cocycles.

For a group $G$ acting on an abelian group $M$, group cohomology consists of the groups $H^n(G,M)$ obtained from the standard cochain complex of maps $G^n\to M$. In degree zero it gives the invariant subgroup $M^G$. In degree one it gives one-cocycles modulo group coboundaries.

If the action is trivial, the one-cocycle identity becomes $f(gh)=f(g)+f(h)$ and all one-coboundaries vanish, so $H^1(G,M)=\mathrm{Hom}(G,M)$. Degree two encodes group extensions with the prescribed action on the kernel. This explains why cohomology naturally measures whether local multiplication data can be assembled into a globally associative extension.

A crossed homomorphism is a map $\varphi :G\to M$ satisfyingFor a one-cochain, the formula in part i givesso the crossed homomorphisms are exactly the one-cocycles. A zero-cochain $m\in M$ has coboundary $g\mapsto gm-m$, the principal crossed homomorphism associated with $m$. Therefore

Write every element as $a^k{b}^l$ with $k\in \{0,1\}$. The action of this element on $M=\mathbb{Z}$ is multiplication by $(-1{)}^k$. A direct check of the four possibilities for the two exponents of $a$ showsso $\varphi$ is a crossed homomorphism and hence a one-cocycle.

It cannot be principal: if $\varphi (g)=gm-m$ for some $m\in \mathbb{Z}$, then at $g=a$ one would havewhich is impossible in the integers. Thus $[\varphi ]\ne 0$ and

For the free presentation $G=F/R$, conjugation in $F$ gives the relation modulewhere $f\in F$ is any lift of $g\in G$. A different lift differs by an element of $R$, whose inner conjugation acts trivially on the abelianization, so this is a well-defined $\mathbb{Z}G$-module action.

Choose free generators $x_1,\dots ,x_n$ of $F$. The presentation relation sequence becomeswhere $d_1(e_i)={\bar{x}}_i-1$. In Fox calculus, the first map isThe Fox identity $r-1={\sum }_i(\partial r/\partial x_i)(x_i-1)$ gives $d_{1}j=0$, and the standard lifting argument in the free group proves exactness.

Split the sequence at the augmentation ideal $I_G$. Applying ${\mathrm{Hom}}_{\mathbb{Z}G}(-,M)$ toand using projectivity of $(\mathbb{Z}G{)}^n$ givesThe other short exact sequence $0\to I_G\to \mathbb{Z}G\to \mathbb{Z}\to 0$ identifies the last group with $H^2(G,M)$. A one-cocycle on $F$ is a derivation and is determined freely by its values on $x_1,\dots ,x_n$, so $H^1(F,M)$ is $M^n$ modulo principal derivations. Principal derivations vanish on $R$, and restriction sends a derivation $d$ to the $G$-map $r[R,R]\mapsto d(r)$. The preceding cokernel sequence therefore descends to the Mac Lane exact sequence for a free presentation

The left map need not be injective. Take $F=⟨t⟩\cong \mathbb{Z}$, $R=⟨t^m⟩$, $G\cong C_m$, and the trivial module $M=\mathbb{Z}/m\mathbb{Z}$, where $m>1$. Then $H^1(F,M)\cong M$, while restriction sends the derivation determined by $d(t)=a$ toThus the left map is zero although its domain is nonzero, and $R$ is nontrivial.

Put $Q=G/H$ and write $M^H$ for the invariant submodule. The five-term exact sequence in group cohomology associated with the Lyndon–Hochschild–Serre spectral sequence isThe inflation map in group cohomology composes a cocycle on $Q$ with the quotient homomorphism $G\to Q$. The restriction map in group cohomology restricts a cocycle from $G$ to $H$. The quotient action on the middle term is, for $q=gH$ and a one-cocycle $f$,This is independent of the lift and of the representative at the level of cohomology. Finally, the transgression in group cohomology extends a $Q$-invariant class on $H$ to a one-cochain on $G$; its coboundary is $H$-basic and descends to the two-cocycle on $Q$ representing $d_2[f]$. Changing the extension changes that cocycle by a group coboundary.

For the application, choose free generators $x_1,\dots ,x_n$ of $F$ and normal generators $r_1,\dots ,r_n$ of $R$. Since a finite nonabelian simple group $K$ is a perfect group, its abelianization is zero. The five-term sequence for $1\to R\to F\to K\to 1$ with trivial coefficients containsThe first term is zero because $K$ is finite, and the last term is zero because a free group has cohomological dimension one. It remains to prove that restriction is surjective.

The invariant submodule of homomorphisms $R\to \mathbb{Z}$ is exactlyThe images of the relators $r_1,\dots ,r_n$ generate $R/[F,R]$, so such a homomorphism $\lambda$ is determined by the integer vector $b=(\lambda (r_1),\dots ,\lambda (r_n){)}^T$. Let $A$ be the relator exponent-sum matrix. This square integer matrix presents $K^{\mathrm{ab}}$, which is zero, so $A$ is a unimodular matrix. There is therefore an integer vector $v$ satisfying $Av=b$. Define $\psi :F\to \mathbb{Z}$ by assigning to $x_i$ the $i$th entry of $v$. The definition of $A$ gives $\psi (r_j)=\lambda (r_j)$ for every $j$. Since the relator images generate $R/[F,R]$, the restriction of $\psi$ to $R$ equals $\lambda$. Restriction is surjective, exactness now gives

For a group extension $1\to H\to G\to Q\to 1$ and a $G$-module $M$, the transgressionis the first spectral sequence differential crossing from the vertical edge to the horizontal edge of the Lyndon–Hochschild–Serre spectral sequence. By the five-term exact sequence in group cohomology, an invariant one-cocycle class on $H$ has zero transgression precisely when it is the restriction of a one-cocycle class on $G$. Thus transgression measures an extension obstruction. An arbitrary extension of a representative to a group cochain on $G$ need not have a coboundary that already descends to $Q$; the cochain construction requires appropriate corrections.

For $1\to C_2\to C_4\to C_2\to 1$ with trivial coefficients ${\mathbb{F}}_2$, every group homomorphism $C_4\to {\mathbb{F}}_2$ kills its order-two subgroup. Consequently the nonzero class in $H^1(C_2,{\mathbb{F}}_2)$ has nonzero transgression. It detects this nonsplit group extension, represented by the carry two-cocycle $c(a,b)=ab$ for $a,b\in \{0,1\}$.