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# 3.3: Subgrupos

• • Tom Denton
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Cuando consideramos las simetrías de, digamos, un pentágono, notamos que tiene simetrías rotacionales como el pentágono “accidentado”. Desde el pentágono accidentado, vemos que las rotaciones mismas forman un grupo; ¡hay un grupo de rotaciones dentro del grupo de simetrías del pentágono! De igual manera, si consideramos solo el flip, vemos un grupo similar a las simetrías de la cara perfectamente simétrica. Esto produce otro grupo dentro de las simetrías del pentágono. Podemos hacer esto preciso:

##### Definición 3.2.0: Subgrupo

Let$$G$$ be a group, and $$H$$ a subset of $$G$$. Then $$H$$ is a subgroup of $$G$$ if $$H$$ is itself a group using the same operation as $$G$$.

Ostensiblemente, para comprobar que un subconjunto$$H$$ is a subgroup, we would need to check all four properties of the group. That is, closure (ie, the operation gives a map $$H\times H\rightarrow H$$; products of things in $$H$$ are always in $$H$$), identity, the existence of inverses, and associativity.

De hecho, ya que$$H$$ has the same operation as $$G$$, we know that the operation in $$H$$ is associative (since $$G$$ is a group). Furthermore, if the operation is closed and inverses exist, then we know that for any $$h\in H$$, $$hh^{-1}=e$$ must be in $$H$$. So really we only need to check two things:

1. Cierre:$$gh\in H$$ for all $$g,h\in H$$, and
2. Inversa:$$h^{-1}\in H$$ for all $$h\in H$$.

Algunas cosas importantes a tener en cuenta:

1. El grupo$$G$$ is always a subgroup of itself! ($$G$$ is a subset of itself, which is a group with the same operation as $$G$$.)
2. ¡El subconjunto que contiene solo el elemento de identidad también es un subgrupo! A esto se le llama el subgrupo trivial.
3. El conjunto de todos los poderes de un elemento$$h$$ ($$\{\ldots, h^{-1}, h^{-2}, e, h, h^2, \ldots\}$$) is a subgroup of $$G$$. This is called the cyclic subgroup generated by $$h$$.
##### Ejercicio 3.2.1

Let$$X$$ be a geometric object. Show that the rotations of $$X$$ back onto itself forms a subgroup of the group of symmetries of $$X$$. (Try this in particular on a regular polygon and a regular polyhedron. What happens with a 'bumpy' polygon?)

Let$$G$$ be a group, and $$g\in G$$. Consider a function $$f_g:G\rightarrow G$$ given by $$f_g(h)=g\cdot h$$. (This is the 'left multiplication by $$g$$' function.) What happens if, for some $$h, k \in G$$, $$f_g(h)=f_g(k)$$? Then $$gh=gk$$, so $$g^{-1}gh=g^{-1}gk$$, and $$h=k$$. This tells us that $$f_g$$ is a one-to-one, or injective, function. If $$G$$ has a finite number of elements, then $$f_g$$ is also an onto function, and is thus a bijection from $$G$$ back to itself. Then we can consider $$f_g$$ as a permutation of $$G$$!

Si consideramos$$G$$ as a set, we can think of any left multiplication as a permutation of $$G$$. But the set of all left multiplications is itself a group. This gives us what is known as Cayley's Theorem!

##### Teorema 3.2.2: Teorema de Cayley

La ley de gas ideal es fácil de recordar y aplicar en la resolución de problemas, siempre y cuando consigas los valores adecuados a

##### Ejercicio 3.2.3

Etiquetar las seis simetrías del triángulo equilátero. Demostrar que las simetrías del triángulo son un subgrupo de$$S_6$$, the permutations of $$6$$ objects.

Vale la pena notar que para cualquier$$g$$ in a group $$G$$, the powers of $$g$$ generate a subgroup of $$G$$. The set $$\{g^i \mid i \in \mathbb{Z} \}$$ is closed under the group operation, and includes the identity and inverses. This is called the cyclic subgroup generated by $$g$$.

##### Ejercicio 3,2,4

Encuentra todos los subgrupos del grupo de permutación$$S_3$$ for three objects. Which subgroups are subgroups of other subgroups? Name each subgroup, and arrange them according to which is contained in which.