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TrickyDicky
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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?
Thanks, I was under the impression that the isometry group of ##S^3## was ##SO(4)##.N1k1tA said:Yes, there are many such examples. if I'm not mistaken [tex]\mathrm{Aut}(S_n)=S_n[/tex] for all [tex]n[/tex] except [tex]2,6[/tex]. But you can check it by hand that [tex]\mathrm{Aut}(S_3)=S_3[/tex]. The groups with this property are called complete groups.
TrickyDicky said:Thanks, I was under the impression that the isometry group of ##S^3## was ##SO(4)##.
Oh, sorry, I should have made clear I referred to continuous spaces and groups.N1k1tA said:##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?
An automorphism group is a mathematical concept that describes a group of transformations that preserve the structure of a mathematical object. In other words, it is a set of transformations that do not change the essential properties of the object.
Some examples of automorphism groups include the group of symmetries of a geometric shape, the group of all rotations in three-dimensional space, and the group of all bijections on a set.
An automorphism space is a mathematical space that contains all possible automorphisms of a given object. It is often used to study the symmetries and properties of the object.
An automorphism group is a specific type of automorphism space that contains all possible transformations that preserve the structure of an object. In other words, it is a subset of the automorphism space.
Automorphism groups and spaces have applications in various fields including geometry, topology, group theory, and cryptography. They are also used in computer science to study the symmetries and transformations of data structures and algorithms.