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Let's say you have a group like $(\mathbb{R}, +)$, where it’s not immediately clear where the symmetries are (compared to a group $(S, \circ)$ which is a group of symmetries of some object equipped with composition).

I'm trying to come up with a procedure to turn any group into another group where the symmetries are more obvious.

On a high level, I can use Cayley's theorem to find a subgroup of a symmetric group that is isomorphic to my original group.

I want to know what happens on a low level. My current hypothesis is that you consider the symmetries of the group itself, put them in a set, and equip these transformations with composition, you get the group I'm looking for. I presume what I have described is Cayley's theorem itself.

This checks out for the group $(\mathbb{R}, +)$, where the symmetries of the group are the translations on $\mathbb{R}$.

Is what I'm saying correct?

Shaun
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Davide Radaelli
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    Yes, Cayley means that multiplication in a group $G$ is also a group action of $G$ on the set $G.$ For some groups, it is more obvious what the symmetries are. For example, $(\mathbb R,+)$ is the collection of permutations $\sigma$ on $\mathbb R$ such that $\sigma(a)-\sigma(b)=a-b$ for all $a,b.$ This can be seen as order-preserving isometries on $\mathbb R.$ Finite groups can always be represented as the symmetries of an edge-colored directed graph, of course. – Thomas Andrews Mar 18 '22 at 20:39
  • You can do the same for infinite groups, but that representation is far from an intuitive object for infinite groups. – Thomas Andrews Mar 18 '22 at 20:41
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    My answer to your previous question describes exactly this. And yes, as I say there this is just Cayley's theorem itself. – Noah Schweber Mar 19 '22 at 02:56
  • @NoahSchweber I guess I'm just shocked that Cayley's theorem is so simple, I thought I'd made a mistake and wanted to double check. – Davide Radaelli Mar 19 '22 at 18:42
  • @DavideRadaelli Nope, it's really that simple. (Like any good simple theorem it has plenty complicated follow-ups, but that's a separate story.) – Noah Schweber Mar 19 '22 at 18:59
  • Incidentally, is there any aspect of this question not addressed by my answer to your previous one? – Noah Schweber Mar 19 '22 at 19:08
  • I didn't make the link that what happens with Cayley's theorem is that you find the symmetries of the group itself, and equip it with composition (and this new group is isomorphic to the old). – Davide Radaelli Mar 19 '22 at 20:44
  • Any object in math can be turned into a symmetry group by considering its symmetries. Even groups themselves. – Davide Radaelli Mar 19 '22 at 20:46
  • Let us continue this discussion in chat. – Davide Radaelli Mar 19 '22 at 20:48

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