Definition (Group action) An action of a group $G$ on a mathematical object $X$ is a group homomorphism $G \rightarrow \mathrm{Sym}(X)$.

i.e. Given an action $f$ of a group $G$ on a mathematical object $X$.

$f : G \times X \rightarrow X$

$f(g,x)=g \cdot x $

The associated group homomorphism $h: G \rightarrow \mathrm{Sym}(X)$ is:

$h(g)=g\cdot X = \{g \cdot x :\forall x\in X\}\in \mathrm{Sym}(X)$

Definition Given a group action of a group $G$ on an object $X$, the homomorphism $G \rightarrow \mathrm{Sym}(X)$ is faithful if it is monomorphism (injective)

Example The dihedral group $D_8$ permutes the vertices of the regular $n$-gon, giving a representation $f:D_8 \rightarrow \mathrm{Sym}(n)$. And $\mathrm{Kernel} = \{g \in D_8 : f(g)=(\space )\}=\{e\}$. So the $f$ is a monomorphism, therefore every element in $D_8$ can be faithfully represented as a group of permutation.

But is it true that every group even the uncountably infinite group such as $(\mathbb{Z},+)$ be faithfully represented as a group of permutation? Shouldn't the group be finite and countable?

  • $\begingroup$ "Shouldn't the group be finite and countable?" First, finite implies countable. Second, why? If you examine the standard argument for how all finite groups have a faithful permutation action (Cayley's theorem), does it assume finiteness anywhere? $\endgroup$ – blue Jul 25 '14 at 8:31
  • $\begingroup$ Also, $gX=\{gx:x\in X\}$ is an orbit, which is a subset of $X$, not an element of ${\rm Sym}(X)$. $\endgroup$ – blue Jul 25 '14 at 8:32
  • $\begingroup$ @blue orbit is $G \cdot x$ $\endgroup$ – MathsMy Jul 25 '14 at 8:33
  • $\begingroup$ @blue I am aware that this is Cayley's theorem, I am not sure about the standard argument for this statement $\endgroup$ – MathsMy Jul 25 '14 at 8:34
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    $\begingroup$ Permutations are bijections $X\to X$ for some set $X$. There are no restrictions on $X$'s size. If $X$ is finite then without loss of generality we can assume $X=\{1,\cdots,n\}$, and this standardization is convenient for computational and pedagogical purposes. (It is quite obvious that there is no faithful finite permutation action of infinite groups; that would amount to an injection from an infinite group to a finite group.) $\endgroup$ – blue Jul 25 '14 at 8:51

Cayley proved that every group, even groups with an uncountable number of elements, can be represented by the permutation group of some underlying set. Maybe this Wikipage can help you: http://en.wikipedia.org/wiki/Cayley_group

  • $\begingroup$ Oh, the permutation group does not necessarily mean $Sym(n)$ right? $\endgroup$ – MathsMy Jul 25 '14 at 8:35
  • $\begingroup$ No. That's the set of permutations of the first n integers. You can have permututations where even an uncoutable number of elements are permutated, e.g. rotations, translations or refelctions in the Euclidian plane. $\endgroup$ – Steven Van Geluwe Jul 25 '14 at 8:40
  • $\begingroup$ So when it says "every group can be faithfully represented as a group of permutations", it just means that every element of a group can be associated with a "bijective function", right? I thought group of permutation is $sym(n)$ but it's group of bijective functions... $\endgroup$ – MathsMy Jul 25 '14 at 8:45
  • $\begingroup$ Even in your definition of a group action you talk about $\mathrm{Sym}(X)$ for arbitrary $X$ ... $\endgroup$ – Martin Brandenburg Jul 25 '14 at 8:47

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