5.2: Ejemplos de grupos de cocientes

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Ahora que hemos aprendido un poco sobre los subgrupos y cocientes normales, deberíamos construir más ejemplos.

Enteros mod$n$, Again

Recordemos al grupo$\mathbb{Z}_n$. ¡Esto también se puede realizar como el grupo cociente!

Let$n\mathbb{Z}$ denote the set of integers divisible by $n$: $n\mathbb{Z}=\{\ldots, -3n, -2n, -n, 0, n, 2n, 3n, \ldots\}$. This forms a subgroup: $0$ is always divisible by $n$, and if $a$ and $b$ are divisible by $n$, then so is $a+b$. Since every subgroup of a commutative group is a normal subgroup, we can from the quotient group $\mathbb{Z}/\mathord n\mathbb{Z}$.

Para ver esto concretamente, vamos$n=3$. Then the cosets of $3\mathbb{Z}$ are $3\mathbb{Z}$, $1+3\mathbb{Z}$, and $2+3\mathbb{Z}$. We can then add cosets, like so: $(1+3\mathbb{Z}) + (2+3\mathbb{Z}) = 3+3\mathbb{Z} = 3\mathbb{Z}.$ The last equality is true because $3\mathbb{Z}=\{\ldots, -6, -3, 0, 3, 6, \ldots\}$, so that $3+3\mathbb{Z}=\{\ldots, -3, 0, 3, 6, 9, \ldots\}=3\mathbb{Z}$.

Escribir tablas de adición para$\mathbb{Z}/\mathord 5\mathbb{Z}$ as a quotient group, and check that it is isomorphic to $\mathbb{Z}_5$ as previously defined.

El Grupo Alternante

Otro ejemplo es un subgrupo muy especial del grupo simétrico llamado el grupo Alternante,$A_n$. Hay un par de formas diferentes de interpretar el grupo alterno, pero principalmente se reduce a la idea del signo de una permutación, que es siempre$\pm 1$. El conjunto$\{1, -1\}$ forma un grupo bajo multiplicación, isomórfico a$\mathbb{Z}_2$. El signo de una permutación es en realidad un homomorfismo. Existen numerosas formas de calcular el signo o una permutación:

Determinantes. Una matriz de permutación es la matriz de la transformación lineal de$n$-dimensional space sending the $i$-th coordinate vector $e_i$ to $e_{\sigma(i)}$. Such matrices have entries all equal to zero or one, with exactly one 1 in each row and each column. One can easily show that such a matrix has determinant equal to $\pm 1$. Since the determinant is a multiplicative function - $\det (MN) = \det(M) \det(N)$ - we can observe the the determinant is a homomorphism from the group of permutation matrices to the group $\{\pm 1\}$.
Contar inversiones. Una inversión en una permutación$\sigma$ is a pair $i<j$ with $\sigma(i)>\sigma(j)$. For example, the permutation $[3,1,4,2]$ has $\sigma(1)>\sigma(2), \sigma(1)>\sigma(3)$ and $\sigma(3)>\sigma(4)$, and thus has three inversions. If there are $i$ inversions, then the sign of the permutation is $(-1)^i$.
Contar cruces. Dibuja una notación trenzada para la permutación donde no se crucen más de dos líneas en ningún punto y ninguna línea se intersecta a sí misma. Luego cuente el número de cruces,$c$. Then $s(\sigma)=(-1)^{c}$. The alternating group is the subgroup of $S_n$ with $s(\sigma)=1$. (To prove that this method of counting works, one needs a notion of Reidemeister moves, which originally arise in the fascinating study of mathematical knots.)

Encuentre el número de inversión para cada permutación en$S_4$, and then sort the permutations by their inversion number.

Demostrar que cada una de las tres definiciones del signo de una permutación dan un homomorfismo de$S_n$ to $\{1, -1\}$. (For the third definition, a sketch of a proof will suffice. Be sure to argue that different braid notations for the same permutation give the same sign, even if the total number of crossings is different.)

Llamamos a una permutación con signo$+1$ a positive permutation, and a permutation with sign $-1$ a negative permutation.

Demostrar que hay$\frac{n!}{2}$ positive permutations in $S_n$.

Ahora podemos definir el grupo alterno.

Definición 5.1.4: Grupos alternos

El grupo alterno$A_n$ is the kernel of the homomorphism $s: S_n\rightarrow \mathbb{Z}_2$. Equivalently, $A_n$ is the subgroup of all positive permutations in $S_n$.

Escribe todos los elementos$A_4$ as a subgroup of $S_4$. Find a nice generating set for $A_4$ and make a Cayley graph.

De hecho, el grupo alterno tiene exactamente dos cosets. El grupo cociente$S_n/\mathord A_n$ is then isomorphic to $\mathbb{Z}_2$.

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Figura 5.1.2: Cociente de

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Escribir tablas de adición para\(\mathbb{Z}/\mathord 5\mathbb{Z}\) as a quotient group, and check that it is isomorphic to \(\mathbb{Z}_5\) as previously defined.

Encuentre el número de inversión para cada permutación en\(S_4\), and then sort the permutations by their inversion number.

Demostrar que hay\(\frac{n!}{2}\) positive permutations in \(S_n\).

Definición 5.1.4: Grupos alternos

Escribe todos los elementos\(A_4\) as a subgroup of \(S_4\). Find a nice generating set for \(A_4\) and make a Cayley graph.

Enteros mod\(n\), Again

Escribir tablas de adición para\(\mathbb{Z}/\mathord 5\mathbb{Z}\) as a quotient group, and check that it is isomorphic to \(\mathbb{Z}_5\) as previously defined.

El Grupo Alternante

Encuentre el número de inversión para cada permutación en\(S_4\), and then sort the permutations by their inversion number.

Demostrar que hay\(\frac{n!}{2}\) positive permutations in \(S_n\).

Definición 5.1.4: Grupos alternos

Escribe todos los elementos\(A_4\) as a subgroup of \(S_4\). Find a nice generating set for \(A_4\) and make a Cayley graph.