Are There Spaces That Are Their Own Automorphisms Group?

  • Context: Graduate 
  • Thread starter Thread starter TrickyDicky
  • Start date Start date
  • Tags Tags
    Groups
Click For Summary

Discussion Overview

The discussion revolves around the concept of spaces that are their own automorphisms group, specifically exploring examples and properties of such spaces. The scope includes theoretical aspects of group theory and topology, with references to specific groups and their automorphisms.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about examples of spaces that are their own automorphisms group.
  • Another participant asserts that for groups ##S_n##, the automorphisms group coincides with the group itself for all ##n## except ##2,6##, and mentions that these groups are referred to as complete groups.
  • A later reply expresses confusion regarding the isometry group of ##S^3##, suggesting it is ##SO(4)##, and clarifies the distinction between ##S^3## (the 3-sphere) and ##S_3## (the symmetric group on three letters).
  • Further clarification is provided about the nature of the groups being discussed, with a participant indicating that their interest pertains to continuous spaces and groups.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between specific groups and their automorphisms, with some asserting examples while others seek clarification on the types of groups being discussed. The discussion remains unresolved regarding the broader implications of these examples.

Contextual Notes

There is a lack of consensus on whether the focus is on Lie groups or arbitrary groups, which may affect the interpretation of the examples provided.

TrickyDicky
Messages
3,507
Reaction score
28
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?
 
Physics news on Phys.org
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.
 
Last edited:
N1k1tA said:
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)##.
 
TrickyDicky said:
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?
 
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?
Oh, sorry, I should have made clear I referred to continuous spaces and groups.
 

Similar threads

Undergrad Are linear automorphisms nothing but the identity mapping?
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Is the Order of an Automorphism in a Field with Characteristic p Equal to p?
  • · Replies 2 ·
Replies
2
Views
2K
MHB 4 problems regarding automorphisms/homomorphisms
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Finding All Automorphisms of Group
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad What is difference between transformations and automorphisms
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Abstract Algebra: Automorphisms
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Groups of Automorphisms - Aut(C) ....
  • · Replies 19 ·
Replies
19
Views
4K
Undergrad Question about "A Group Epimorphism is Surjective"
  • · Replies 13 ·
Replies
13
Views
2K
Graduate What is a free product of groups or vector space?
  • · Replies 2 ·
Replies
2
Views
2K
MHB Understanding Aut(G) Isomorphism to Z2
  • · Replies 34 ·
2
Replies
34
Views
7K