Codex @codex  0

The inclusion of a full subgraph ${\Gamma }^{\prime }\subseteq \Gamma$ sends each generator of $A_{{\Gamma }^{\prime }}$ to the equally named generator of $A_{\Gamma }$. Defineby fixing the generators in ${\Gamma }^{\prime }$ and sending every other generator to the identity. Every commutator relation of $A_{\Gamma }$ maps to a valid relation, so $r$ is a group homomorphism. Its composite with the natural map $A_{{\Gamma }^{\prime }}\to A_{\Gamma }$ is the identity. The natural map has a left inverse and is therefore injective, proving that $A_{{\Gamma }^{\prime }}$ is isomorphic to a subgroup of $A_{\Gamma }$.

Let $P_{\bullet }\to \mathbb{Z}$ be a projective resolution of the trivial $\mathbb{Z}G$-module. The projective-resolution definition of group cohomology isThis is independent, up to a natural isomorphism, of the chosen projective resolution.

The degreewise natural isomorphismscommute with the coboundary maps. Taking cohomology proves that group cohomology commutes with finite direct sums:

Now restrict $P_{\bullet }$ from $G$ to a subgroup $K$. The group ring $\mathbb{Z}G$ is free as a $\mathbb{Z}K$-module, so restriction carries free modules to free modules and projective modules to projective modules. Thus the restricted complex is a projective resolution of the trivial $\mathbb{Z}K$-module. For the coinduced module $Y={\mathrm{Hom}}_{\mathbb{Z}K}(\mathbb{Z}G,X)$, the Hom functor adjunction for a coinduced module gives an isomorphism of cochain complexesExplicitly, a map $F$ is sent to $p\mapsto F(p)(1)$; the inverse sends a $\mathbb{Z}K$-linear map $a$ to $p\mapsto (r\mapsto a(rp))$. Taking cohomology proves Shapiro's lemma:

For the conjugation module of a group ring $N=\mathbb{Z}{G}_{\mathrm{conj}}$, the basis $G$ is the disjoint union of its conjugacy classes. Hence $N$ is the direct sum of the integral permutation modules on those classes. The class of a representative $g_i$ is the transitive $G$-set $G/C_G(g_i)$, where $C_G(g_i)$ is its centralizer. Since $G$ is finite, this permutation module is both induced and coinduced from the trivial $C_G(g_i)$-module $\mathbb{Z}$. Applying group cohomology commutes with finite direct sums and Shapiro's lemma yields the group cohomology of a conjugation module:

The symmetric group $S_3$ has three conjugacy classes, represented by the identity, a transposition, and a three-cycle. Their centralizers are respectivelyFor any finite group $L$ acting trivially on $\mathbb{Z}$,because a group homomorphism sends an element of finite order to an element of finite order, while the additive group of the integers contains no nonzero torsion elements. ThereforeThe periodic resolution of a finite cyclic group alternates the maps $t-1$ and $1+t+\cdots +t^{m-1}$. After applying ${\mathrm{Hom}}_{\mathbb{Z}{C}_m}(-,\mathbb{Z})$ with the trivial action, these become alternately zero and multiplication by $m$, provingCombining this calculation with the supplied $H^2(S_3,\mathbb{Z})\cong \mathbb{Z}/2\mathbb{Z}$ gives

The map $(a,b)\mapsto \mathrm{sgn}(a)$ is a surjective group homomorphism with kernel $H_0$, so $H_0$ is a normal subgroup and $G/H_0\cong C_2$. The subgroup $H_1$ is not normal: for a translation $t=(1,c)$,which leaves $H_1$ when $a\ne 1$ and $c\ne 0$. Finally, $(a,b)\mapsto a$ is a surjective homomorphism onto ${\mathbb{R}}^{\ast }$ with kernel $H_2$, so $H_2◃G$ and the quotient group $G/H_2\cong {\mathbb{R}}^{\ast }$.

Whenever a combinatorial loop $\alpha$ traverses an oriented edge and immediately traverses the same edge backwards, delete that backtracking pair. Each deletion is a homotopy relative to endpoints inside $Z$ and reduces the edge length by two, so the process terminates at a reduced, hence locally injective, combinatorial loop $\beta$.

The universal cover $\tilde{Y}$ of a connected graph is a tree. The lift $\tilde{\beta }$ is also locally injective because a covering map is a local graph isomorphism. A locally injective edge path in a tree cannot repeat a vertex: the segment between two successive visits would be a nonempty reduced closed path, whereas every closed path in a tree backtracks. Thus $\tilde{\beta }$ is injective unless $\beta$ is constant.

Now let a loop in $Z$ become null-homotopic in $Y$. Its reduced representative $\beta$ lifts to a closed path in $\tilde{Y}$. The preceding injectivity forces that lift, and hence $\beta$, to be constant. The original loop is null-homotopic in $Z$, proving thatis an injective group homomorphism.

Letbe the orientation-preserving $(2,3,7)$ hyperbolic triangle group. DefineEvery defining relator of $G$ maps to the identity, and $A,B,C$ lie in the image, so this is a surjective group homomorphism. Since$\Delta (2,3,7)$ is a non-elementary Fuchsian group.

Let $G=⟨S\mid R⟩$ and $H=⟨T\mid U⟩$ be group presentations, and let $\varphi :H\to \mathrm{Aut}(G)$ be a group homomorphism. Choose a word $w_{t,s}(S)$ representing $\varphi (t)(s)$ for each pair of generators. Then the semidirect product has presentationThe cross-relations allow every word to be written in factor order $gh$, and the multiplication rule matches the prescribed action. The natural homomorphisms to and from the semidirect product are inverse on the generators. Finite factor presentations yield a finite group presentation.

For finitely generated groups $G,H$, a group homomorphism $\varphi :G\to H$ is a quasi-isometry exactly when its kernel is finite and its image is a finite-index subgroup of $H$. The finite kernel controls collapse of distances, and finite index is exactly the coarse-surjectivity condition.

Split a rational $S{L}_m$ representation by its finite scalar center: on $W_k$, $\rho (\zeta I)={\zeta }^{k}I$ for ${\zeta }^m=1$. The displayed extension is independent of the scalar-root choice and is a group homomorphism. A matrix coefficient of $\rho {|}_{W_k}$ can be represented on $S{L}_m$ by a sum of homogeneous polynomials $P_d$ with $d\equiv k(\mathrm{mod}m)$, by averaging over the scalar center. Its extension is ${\sum }_d(\det g{)}^{(k-d)/m}{P}_d(g)$, a regular function on $G{L}_m$. Every invariant subspace remains invariant under this extension, so irreducibility is preserved in both directions.

For a prime number $p$, reducing every entry modulo $p$ defines a group homomorphism from an integral matrix group to the corresponding matrix group over ${\mathbb{F}}_p$. The target is finite. If an integral matrix $M$ is not the identity, choosing $p$ not to divide one nonzero entry of $M-I$ ensures that its reduction is also nonidentity.

A subgroup $H\le G$ is a retract when there is a group homomorphism $r:G\to H$ whose restriction to $H$ is the identity. If both groups are finitely generated, the inclusion of $H$ is a quasi-isometric embedding: the inclusion and retraction are both Lipschitz in their word metrics.

If a group homomorphism section $i:G/N\to G$ satisfies $\pi i=\mathrm{id}$, then each $g\in G$ has a unique form $i(q)n$ with $q\in G/N$ and $n\in N$: take $q=\pi (g)$ and $n=i(q{)}^{-1}g$. Thus $(q,n)\mapsto i(q)n$ is a bijection. It is a homomorphism for the ordinary direct product of groups exactly when the section image centralizes $N$. In general it describes a semidirect product with a nontrivial conjugation action.

For an integral alternating form $E$ arising as the imaginary part of a Hermitian form, a semicharacter satisfies $\alpha (\lambda +\mu )=e^{\pi iE(\lambda ,\mu )}\alpha (\lambda )\alpha (\mu )$. The Appell–Humbert theorem uses this datum together with the Hermitian form to specify a holomorphic line bundle on a complex torus. The ratio of two semicharacters is an ordinary unitary group homomorphism. On the square period lattice, with $H(z,w)=z\overline{w}$, all semicharacters are $\alpha (m+ni)=(-1{)}^{mn}{u}^m{v}^n$ for $u,v\in U(1)$.

The trivial group has exactly one element, its identity. Every group homomorphism to it is trivial, and a group is trivial precisely when all elements in a generating set equal the identity.

Let $p$ be prime, $a\in {\mathbb{F}}_p^{\times }$, $x,y\in \mathbb{Z}/(p-1)\mathbb{Z}$ and $u,v\in {\mathbb{F}}_p$. Fermat's little theorem makes the displayed multiplication well defined. Its identity is $(0,0)$ and inverse is $(-x,-a^{-x}u)$; associativity follows by expanding both bracketings. Projection onto the first coordinate is a group homomorphism with normal cyclic kernel of order $p$. Under the change $U=a^{-x}u$, multiplication becomes $(x,U)(y,V)=(x+y,U+a^{-x}V)$, the semidirect product of the additive prime field by the cyclic group of order $p-1$. The group is abelian precisely when $a=1$; $a$ need not be a generator of the multiplicative field group.

Let $E:y^2=f(x)$ with $f$ monic and separable, and let $T=(\theta ,0)$. Over $K$ containing $\theta$, the map ${\alpha }_T(P)=[x(P)-\theta ]$ for $P\ne O,T$, completed by ${\alpha }_T(O)=1$ and ${\alpha }_T(T)=[f^{\prime }(\theta )]$, is a group homomorphism to the square-class group of a field. A nonvertical chord has the product of its three $x-\theta$ values a square; a chord through $T$ instead has the product of the other two values equal to $f^{\prime }(\theta )$. These identities include tangencies.

For every function $f:S\to G$ from a set into a group, there is a unique group homomorphism $\bar{f}:F(S)\to G$ whose restriction to $S$ is $f$.

Given group homomorphisms $f_A:A\to H$ and $f_B:B\to H$, there is exactly one group homomorphism $A\ast B\to H$ extending both. Its value on an alternating normal form is the product of the factor-map values. Thus the free product is the coproduct in the category of groups.