Group elements as permutation of a set

Question

I'm studying deeply, starting from different POVs, group actions, stabilizers, etc. Quoting from Rotman's Theory of Groups, p.49:

"We have encoutered several situations in which elements of a group may be regarded as permutations of a set."

Actually I was wondering why this is true just for "several situations" and not for "all situations". Try to be more precise with my doubt.

Actually Cayley stated "every group G is isomorphic to a subgroup of a given symmetric group acting on G", so roughly speaking "everything is a permutation group (where "is" is up to isomorphisms, of course).

More than this, a group action can be seen as a homorhism between a group and the Sym(X), i.e. the group of permutation on the set X (having X coinciding with G itself, eventually).

Of course, the action can be faithful or not, but my feeling every element of every group can be seen as a permutation of something.

Am I right in stating so? I guess this is also the same view Galois & Co had initially, before introducing "modern" abstract group", where every group was seen as a permutation of something.

May you help where my sentence could be eventually wrong, please?

thanks in advance

Ricky

Answer 1

It's true that every group can be seen as a permutation of SOMEthing.

But the case we're often interested in is where we have some object $X$, and seek a group $G$ that acts on $X$. An example is where $X$ is a surface, with the group being the group of isometries of the surface. Another example is the set of all possible configurations of a Rubik's cube, with the group being the "operations" one can perform on a cube.

If I give you a Rubik's cube and the group $\Bbb Z/37\Bbb Z$, you're not going to be able to make the latter act on the former (except trivially) --- every element of the group corresponds to "do nothing to the cube."

So the interest isn't "goes a group act on something?" but rather, "When I've got a group acting (nontrivially) on a set $X$, what can I say in general?"

In particular, you might be able to say that the group is not very large (e.g., there are no nontrivial self-isometries of this particular surface), or that it's abelian, or that it has certain subgroups, etc.

To answer your explicit question about what Rotman is saying, let me rephrase. I think he means

We've seen several $(G, X)$ pairs in which $G$ can be said to be acting on $X$,

rather than

We've seen several situations in which for some group $G$, there's a set $X$ with $G$ acting on $X$,

since the latter is (as you observe) always true if we pick $X = G$ and use the action by left-multiplication, say.