Teoremas de isomorfismo

Dejar${\displaystyle G}$y${\displaystyle H}$sean grupos, y dejen${\displaystyle f:G\rightarrow H}$Sea un homomorfismo . Entonces:

1. El núcleo de${\displaystyle f}$es un subgrupo normal de${\displaystyle G}$,
2. La imagen de${\displaystyle f}$es un subgrupo de${\displaystyle H}$, y
3. La imagen de${\displaystyle f}$es isomorfo al grupo cociente${\displaystyle G/\ker f}$.

En particular, si${\displaystyle f}$es sobreyectiva entonces${\displaystyle H}$es isomorfo a${\displaystyle G/\ker f}$.

Este teorema se suele denominar primer teorema de isomorfismo .

Dejar${\displaystyle G}$ser un grupo. Dejemos${\displaystyle S}$ser un subgrupo de${\displaystyle G}$y dejar${\displaystyle N}$ser un subgrupo normal de${\displaystyle G}$. Entonces se cumple lo siguiente:

1. El producto${\displaystyle SN}$es un subgrupo de${\displaystyle G}$,
2. El subgrupo${\displaystyle N}$es un subgrupo normal de${\displaystyle SN}$,
3. La intersección${\displaystyle S\cap N}$es un subgrupo normal de${\displaystyle S}$, y
4. Los grupos de cociente${\displaystyle (SN)/N}$y${\displaystyle S/(S\cap N)}$son isomorfos.

Técnicamente, no es necesario${\displaystyle N}$ser un subgrupo normal, siempre y cuando${\displaystyle S}$es un subgrupo del normalizador de${\displaystyle N}$en${\displaystyle G}$. En este caso,${\displaystyle N}$no es un subgrupo normal de${\displaystyle G}$, pero${\displaystyle N}$sigue siendo un subgrupo normal del producto${\displaystyle SN}$.

This theorem is sometimes called the second isomorphism theorem,^[3]diamond theorem^[4] or the parallelogram theorem.^[5]

An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting ${\displaystyle G=\operatorname {GL} _{2}(\mathbb {C} )}$, the group of invertible 2 × 2 complexmatrices, ${\displaystyle S=\operatorname {SL} _{2}(\mathbb {C} )}$, the subgroup of determinant 1 matrices, and ${\displaystyle N}$ the normal subgroup of scalar matrices ${\displaystyle \mathbb {C} ^{\times }\!I=\left\{\left({\begin{smallmatrix}a&0\\0&a\end{smallmatrix}}\right):a\in \mathbb {C} ^{\times }\right\}}$, we have ${\displaystyle S\cap N=\{\pm I\}}$, where ${\displaystyle I}$ is the identity matrix, and ${\displaystyle SN=\operatorname {GL} _{2}(\mathbb {C} )}$. Then the second isomorphism theorem states that:

${\displaystyle \operatorname {PGL} _{2}(\mathbb {C} ):=\operatorname {GL} _{2}\left(\mathbb {C} )/(\mathbb {C} ^{\times }\!I\right)\cong \operatorname {SL} _{2}(\mathbb {C} )/\{\pm I\}=:\operatorname {PSL} _{2}(\mathbb {C} )}$

Let ${\displaystyle G}$ be a group, and ${\displaystyle N}$ a normal subgroup of ${\displaystyle G}$. Then

1. If ${\displaystyle K}$ is a subgroup of ${\displaystyle G}$ such that ${\displaystyle N\subseteq K\subseteq G}$, then ${\displaystyle G/N}$ has a subgroup isomorphic to ${\displaystyle K/N}$.
2. Every subgroup of ${\displaystyle G/N}$ is of the form ${\displaystyle K/N}$ for some subgroup ${\displaystyle K}$ of ${\displaystyle G}$ such that ${\displaystyle N\subseteq K\subseteq G}$.
3. If ${\displaystyle K}$ is a normal subgroup of ${\displaystyle G}$ such that ${\displaystyle N\subseteq K\subseteq G}$, then ${\displaystyle G/N}$ has a normal subgroup isomorphic to ${\displaystyle K/N}$.
4. Every normal subgroup of ${\displaystyle G/N}$ is of the form ${\displaystyle K/N}$ for some normal subgroup ${\displaystyle K}$ of ${\displaystyle G}$ such that ${\displaystyle N\subseteq K\subseteq G}$.
5. If ${\displaystyle K}$ is a normal subgroup of ${\displaystyle G}$ such that ${\displaystyle N\subseteq K\subseteq G}$, then the quotient group ${\displaystyle (G/N)/(K/N)}$ is isomorphic to ${\displaystyle G/K}$.

The last statement is sometimes referred to as the third isomorphism theorem. The first four statements are often subsumed under Theorem D below, and referred to as the lattice theorem, correspondence theorem, or fourth isomorphism theorem.

Let ${\displaystyle G}$ be a group, and ${\displaystyle N}$ a normal subgroup of ${\displaystyle G}$. The canonical projection homomorphism ${\displaystyle G\rightarrow G/N}$ defines a bijective correspondence between the set of subgroups of ${\displaystyle G}$ containing ${\displaystyle N}$ and the set of (all) subgroups of ${\displaystyle G/N}$. Under this correspondence normal subgroups correspond to normal subgroups.

This theorem is sometimes called the correspondence theorem, the lattice theorem, and the fourth isomorphism theorem.

The Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.^[6]

The first isomorphism theorem can be expressed in category theoretical language by saying that the category of groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This is captured in the commutative diagram in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism ${\displaystyle f:G\rightarrow H}$. The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense; the arbitrary morphism f factors into ${\displaystyle \iota \circ \pi }$, where ι is a monomorphism and π is an epimorphism (in a conormal category, all epimorphisms are normal). This is represented in the diagram by an object ${\displaystyle \ker f}$ and a monomorphism ${\displaystyle \kappa$ :\ker f\rightarrow G} (kernels are always monomorphisms), which complete the short exact sequence running from the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from having to draw the zero morphisms from ${\displaystyle \ker f}$ to ${\displaystyle H}$ and ${\displaystyle G/\ker f}$.

If the sequence is right split (i.e., there is a morphism σ that maps ${\displaystyle G/\operatorname {ker} f}$ to a π-preimage of itself), then G is the semidirect product of the normal subgroup ${\displaystyle \operatorname {im} \kappa }$ and the subgroup ${\displaystyle \operatorname {im} \sigma }$. If it is left split (i.e., there exists some ${\displaystyle \rho :G\rightarrow \operatorname {ker} f}$ such that ${\displaystyle \rho \circ \kappa =\operatorname {id} _{{\text{ker}}f}}$), then it must also be right split, and ${\displaystyle \operatorname {im} \kappa \times \operatorname {im} \sigma }$ is a direct product decomposition of G. In general, the existence of a right split does not imply the existence of a left split; but in an abelian category (such as that of abelian groups), left splits and right splits are equivalent by the splitting lemma, and a right split is sufficient to produce a direct sum decomposition ${\displaystyle \operatorname {im} \kappa \oplus \operatorname {im} \sigma }$. In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence ${\displaystyle 0\rightarrow G/\operatorname {ker} f\rightarrow H\rightarrow \operatorname {coker} f\rightarrow 0}$.

In the second isomorphism theorem, the product SN is the join of S and N in the lattice of subgroups of G, while the intersection S ∩ N is the meet.

The third isomorphism theorem is generalized by the nine lemma to abelian categories and more general maps between objects.

Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules.

Es menos común incluir el Teorema D, generalmente conocido como teorema de retículo o teorema de correspondencia , como uno de los teoremas de isomorfismo, pero cuando se incluye, es el último.

Dejar${\displaystyle R}$y${\displaystyle S}$sean anillos, y dejen${\displaystyle \varphi :R\rightarrow S}$Sea un homomorfismo de anillos . Entonces:

1. El núcleo de${\displaystyle \varphi }$es un ideal de dos caras${\displaystyle R}$,
2. La imagen de${\displaystyle \varphi }$es un subanillo de${\displaystyle S}$, y
3. La imagen de${\displaystyle \varphi }$es isomorfo al anillo cociente${\displaystyle R/\ker \varphi }$.

En particular, si${\displaystyle \varphi }$es sobreyectiva entonces${\displaystyle S}$es isomorfo a${\displaystyle R/\ker \varphi }$. ^{[ 17 ]}

Dejar${\displaystyle R}$ser un anillo. Deja${\displaystyle S}$ser un subanillo de${\displaystyle R}$y dejar${\displaystyle I}$ser un ideal de dos caras${\displaystyle R}$. Entonces:

1. La suma${\displaystyle S+I=\{s+i\mid s\in S,i\in I\}}$es un subanillo de${\displaystyle R}$,
2. La intersección${\displaystyle S\cap I}$es un ideal de dos caras${\displaystyle S}$, y
3. Los anillos de cociente${\displaystyle (S+I)/I}$y${\displaystyle S/(S\cap I)}$son isomorfos.

Dejar${\displaystyle R}$ser un anillo, y${\displaystyle I}$un ideal de dos caras${\displaystyle R}$. Entonces

1. Si${\displaystyle A}$es un subanillo de${\displaystyle R}$de tal manera que${\displaystyle I\subseteq A\subseteq R}$, entonces${\displaystyle A/I}$es un subanillo de${\displaystyle R/I}$.
2. Cada subanillo de${\displaystyle R/I}$es de la forma${\displaystyle A/I}$para algún anillo suba${\displaystyle A}$de${\displaystyle R}$de tal manera que${\displaystyle I\subseteq A\subseteq R}$.
3. Si${\displaystyle J}$es un ideal de dos caras${\displaystyle R}$de tal manera que${\displaystyle I\subseteq J\subseteq R}$, entonces${\displaystyle J/I}$es un ideal de dos caras${\displaystyle R/I}$.
4. Todo ideal de izquierda, derecha o de dos lados de${\displaystyle R/I}$es de la forma${\displaystyle J/I}$para algún ideal de izquierda, derecha o de dos lados${\displaystyle J}$de${\displaystyle R}$de tal manera que${\displaystyle I\subseteq J\subseteq R}$.
5. Si${\displaystyle J}$es un ideal de dos caras${\displaystyle R}$de tal manera que${\displaystyle I\subseteq J\subseteq R}$, entonces el anillo cociente${\displaystyle (R/I)/(J/I)}$es isomorfo a${\displaystyle R/J}$.

Dejar${\displaystyle I}$ser un ideal de dos caras${\displaystyle R}$La correspondencia${\displaystyle A\leftrightarrow A/I}$es una biyección que preserva la inclusión entre el conjunto de subanillos${\displaystyle A}$de${\displaystyle R}$que contienen${\displaystyle I}$y el conjunto de subanillos de ⁠${\displaystyle R/I}$⁠ . ^{[ 18 ]}

Dejar${\displaystyle M}$y${\displaystyle N}$sean módulos, y deje que${\displaystyle \varphi :M\rightarrow N}$Sea un homomorfismo de módulos . Entonces:

1. El núcleo de${\displaystyle \varphi }$es un submódulo de${\displaystyle M}$,
2. La imagen de${\displaystyle \varphi }$es un submódulo de${\displaystyle N}$, y
3. La imagen de${\displaystyle \varphi }$es isomorfo al módulo cociente${\displaystyle M/\ker \varphi }$.

En particular, si${\displaystyle \varphi }$es sobreyectiva entonces${\displaystyle N}$es isomorfo a${\displaystyle M/\ker \varphi }$.

Dejar${\displaystyle M}$ser un módulo, y dejar${\displaystyle S}$y${\displaystyle T}$ser submódulos de${\displaystyle M}$. Entonces:

1. La suma${\displaystyle S+T=\{s+t\mid s\in S,t\in T\}}$es un submódulo de${\displaystyle M}$,
2. La intersección${\displaystyle S\cap T}$es un submódulo de${\displaystyle M}$, y
3. Los módulos de cociente${\displaystyle (S+T)/T}$y${\displaystyle S/(S\cap T)}$son isomorfos.

Sea M un módulo, T un submódulo de M.

1. Si${\displaystyle S}$es un submódulo de${\displaystyle M}$de tal manera que${\displaystyle T\subseteq S\subseteq M}$, entonces${\displaystyle S/T}$es un submódulo de${\displaystyle M/T}$.
2. Cada submódulo de${\displaystyle M/T}$es de la forma${\displaystyle S/T}$para algún submódulo${\displaystyle S}$de${\displaystyle M}$de tal manera que${\displaystyle T\subseteq S\subseteq M}$.
3. Si${\displaystyle S}$es un submódulo de${\displaystyle M}$de tal manera que${\displaystyle T\subseteq S\subseteq M}$, entonces el módulo cociente${\displaystyle (M/T)/(S/T)}$es isomorfo a${\displaystyle M/S}$.

Dejar${\displaystyle M}$ser un módulo,${\displaystyle N}$un submódulo de${\displaystyle M}$. Existe una biyección entre los submódulos de${\displaystyle M}$que contienen${\displaystyle N}$y los submódulos de${\displaystyle M/N}$La correspondencia se da por${\displaystyle A\leftrightarrow A/N}$a pesar de${\displaystyle A\supseteq N}$. Esta correspondencia conmuta con los procesos de tomar sumas e intersecciones (es decir, es un isomorfismo reticular entre el retículo de submódulos de${\displaystyle M/N}$y la red de submódulos de${\displaystyle M}$que contienen${\displaystyle N}$). ^{[ 19 ]}

Dejar${\displaystyle f:A\rightarrow B}$sea ​​un homomorfismo de álgebra . Entonces la imagen de${\displaystyle f}$es una subálgebra de${\displaystyle B}$, la relación dada por${\displaystyle \Phi :f(x)=f(y)}$(es decir, el núcleo de${\displaystyle f}$) es una congruencia en${\displaystyle A}$y las álgebras${\displaystyle A/\Phi }$y${\displaystyle \operatorname {im} f}$son isomorfos . (Nótese que en el caso de un grupo,${\displaystyle f(x)=f(y)}$si y solo si${\displaystyle f(xy^{-1})=1}$(Así pues, se recupera la noción de núcleo utilizada en la teoría de grupos en este caso).

Dado un álgebra${\displaystyle A}$, una subálgebra${\displaystyle B}$de${\displaystyle A}$y una congruencia${\displaystyle \Phi }$en${\displaystyle A}$, dejar${\displaystyle \Phi _{B}=\Phi \cap (B\times B)}$ser el rastro de${\displaystyle \Phi }$en${\displaystyle B}$y${\displaystyle [B]^{\Phi }=\{K\in A/\Phi :K\cap B\neq \emptyset \}}$la colección de clases de equivalencia que se intersecan${\displaystyle B}$. Entonces

1. ${\displaystyle \Phi _{B}}$es una congruencia en${\displaystyle B}$,
2. ${\displaystyle \ [B]^{\Phi }}$es una subálgebra de${\displaystyle A/\Phi }$, y
3. el álgebra${\displaystyle [B]^{\Phi }}$es isomorfo al álgebra${\displaystyle B/\Phi _{B}}$.

Dejar${\displaystyle A}$ser un álgebra y${\displaystyle \Phi ,\Psi }$dos relaciones de congruencia en${\displaystyle A}$de tal manera que${\displaystyle \Psi \subseteq \Phi }$. Entonces${\displaystyle \Phi /\Psi =\{([a']_{\Psi },[a'']_{\Psi }):(a',a'')\in \Phi \}=[\ ]_{\Psi }\circ \Phi \circ [\ ]_{\Psi }^{-1}}$es una congruencia en${\displaystyle A/\Psi }$, y${\displaystyle A/\Phi }$es isomorfo a${\displaystyle (A/\Psi )/(\Phi /\Psi ).}$

Dejar${\displaystyle A}$sea ​​un álgebra y denotemos${\displaystyle \operatorname {Con} A}$el conjunto de todas las congruencias en${\displaystyle A}$. El conjunto ${\displaystyle \operatorname {Con} A}$es una red completa ordenada por inclusión. ^{[ 20 ]} Si${\displaystyle \Phi \in \operatorname {Con} A}$es una congruencia y la denotamos por${\displaystyle \left[\Phi ,A\times A\right]\subseteq \operatorname {Con} A}$el conjunto de todas las congruencias que contienen${\displaystyle \Phi }$(es decir${\displaystyle \left[\Phi ,A\times A\right]}$es un filtro principal en${\displaystyle \operatorname {Con} A}$, además es una subred), entonces el mapa ${\displaystyle \alpha$ :\left[\Phi ,A\times A\right]\to \operatorname {Con} (A/\Phi ),\Psi \mapsto \Psi /\Phi } es un isomorfismo reticular. ^{[ 21 ] [ 22 ]}