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# 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 $$3+3\mathbb{Z}=\{\ldots, -3, 0, 3, 6, 9, \ldots\}=3\mathbb{Z}$$.

## 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:

1. 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 $$\{\pm 1\}$$.
2. 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 $$(-1)^i$$.
3. 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.)
##### 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 $$\mathbb{Z}_2$$.

This page titled 5.2: Ejemplos de grupos de cocientes is shared under a not declared license and was authored, remixed, and/or curated by Tom Denton.