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