Timeline for Isomorphism of Stable groups formed by different sequences
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Sep 8, 2016 at 21:33 | comment | added | Igor Khavkine | @user101496, so you have already have a solution for checking injectivity. I'm not sure that whether you are still missing anything (other than perhaps confidence with the subject). To build up more intuition/confidence I suggest that you look at more examples of inductive limits, of which this is just a special case. | |
| Sep 8, 2016 at 16:18 | comment | added | user101496 | note that drag(A,1) should be diag(A,1) | |
| Sep 8, 2016 at 16:11 | comment | added | user101496 | Thank you again Igor for your explanation. I agree with you so there is no problem with the surjectivity in this case. For the injection: I think we need to define the given maps on equivalence classes otherwise we might get more than one preimage of some elements in G(\infty). For instance the preimage of drag(A,1)\in GL(3) is A and drag(A,1), but if we think that A and drag(A,1) in the same class, then there is no problem anymore. If I am thinking wrong please let me know. Thank you again for your time to explain my questions with diagrams. That might take some time to type them. | |
| Sep 8, 2016 at 14:54 | comment | added | Igor Khavkine | @user101496, there is no requirement that each of the maps in the $p_{2k+1}$, $i_{2k+1,2k}\circ p_{2k}$ sequence to be surjective for the corresponding map $p\colon G(\infty) \to H(\infty)$ to be surjective. Just pick any element $h\in H(\infty)$ and check that there exists a pre-image $g\in G(\infty)$ such that $p(g) = h$. | |
| Sep 8, 2016 at 13:50 | comment | added | user101496 | Thank you so much for your prompt answer. I was thinking the diagrams for a while and I definitely agree with you that i_{3} and i_{5} are isomorphisms of groups and similarly p_{3} and p_{5} too, but if the inclusion mao GL(2) to GL(3) is i_{2,3}, then why i_{2,3} compose p_{2} is surgective? Thank you in advance. | |
| Sep 8, 2016 at 12:34 | history | answered | Igor Khavkine | CC BY-SA 3.0 |