Timeline for Does every automorphism of G come from an inner automorphism of S_G?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Jun 11, 2010 at 19:37 | comment | added | Simon Thomas | en.wikipedia.org/wiki/Hall%27s_universal_group | |
| Jun 11, 2010 at 19:36 | comment | added | Jim Humphreys |
Outer automorphisms are much more mysterious than inner ones: look at the symmetric group $G=S_6$, for instance. You get no practical help in studying its outer automorphisms just by knowing the general fact you quoted. Here the left or right Cayley embedding involves an enormous symmetric group.
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| Jun 11, 2010 at 19:35 | comment | added | James D. Taylor | That sounds interesting. Do you have a reference? | |
| Jun 11, 2010 at 19:34 | comment | added | Simon Thomas | This observation is the basis of a famous construction of Hall. If you iteratively form the embeddings: $G_{0} \to G_{1} \to \cdots \to G_{n} \to \cdots$ where $G_{0} = Sym(5)$ and $G_{n} \to G_{n+1} = Sym(G_{n})$, then the union is a simple locally finite group $G$ such that: (a) every finite group embeds in $G$; and (b) every isomorphism between finite subgroups is induced by an inner automorphism of $G$. | |
| Jun 11, 2010 at 19:27 | comment | added | James D. Taylor | You would think this would be a more famous fact. I was always under the impression that outer automorphisms are much more mysterious than inner automorphisms. | |
| Jun 11, 2010 at 19:26 | vote | accept | James D. Taylor | ||
| Jun 11, 2010 at 19:24 | history | answered | Simon Thomas | CC BY-SA 2.5 |