Are There Spaces That Are Their Own Automorphisms Group?

[TrickyDicky]

Is there any example of an automorphisms group of a space that coincides with the space, i.e. a space that is its own automorphisms group?

 

[N1k1tA]

Yes, there are many such examples. if I'm not mistaken $\mathrm{Aut}(S_n)=S_n$ for all $n$ except $2,6$. But you can check it by hand that $\mathrm{Aut}(S_3)=S_3$. The groups with this property are called complete groups.

 

[TrickyDicky]

Yes, there are many such examples. if I'm not mistaken \mathrm{Aut}(S_n)=S_n for all n except 2,6. But you can check it by hand that \mathrm{Aut}(S_3)=S_3. The groups with this property are called complete groups.

Thanks, I was under the impression that the isometry group of $S^3$ was $SO(4)$.

 

[N1k1tA]

Thanks, I was under the impression that the isometry group of $S^3$ was $SO(4)$.

$S^3$ is the $3$-sphere (the $3$-dimensional sphere embedded in $\mathbb{R}^4$), while $S_3$ is the group of permutations on $3$ letters. Totally different things.

Does your problem concern Lie groups or any arbitrary groups?

 

[TrickyDicky]

$S^3$ is the $3$-sphere (the $3$-dimensional sphere embedded in $\mathbb{R}^4$), while $S_3$ is the group of permutations on $3$ letters. Totally different things.

Does your problem concern Lie groups or any arbitrary groups?

Oh, sorry, I should have made clear I referred to continuous spaces and groups.