3.3: Subgrupos

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.

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