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I am reading Adhikari and Adhikari's (A&A) book, "Basic Modern Algebra with Applications".
I am currently focussed on Section 9.7 Exact Sequences.
On page 391 A&A state and prove the Short Five Lemma. I need help with some of the details of the 'diagram chasing' in the proof.
The Short Five Lemma and the first part of its proof read as follows:https://www.physicsforums.com/attachments/3628
https://www.physicsforums.com/attachments/3629In the first two lines of the above proof we find the following:" ... Suppose for some . We shall show that .
Now since is a monomorphism ... ... "
Now I need someone to critique my detailed reasoning reasoning concerning these statements - I think I understand ... but then, I am working by myself on this material ... so a confirmation that I am on the right track would be most helpful ...Now ... my reasoning is as follows:
We suppose that
We need to show that ... ...
... which implies that ... ...
... which implies that is an injective homomorphism ... ... that is a monomorphism ...Now by the commutativity of the diagram (Fig. 9.7)But since is a homomorphism ...So we have But then is a monomorphism, so that the only element in its domain that gives is ...So then we have ...
... ... and then the proof of (i) continues ...
Can someone please confirm that the details of my analysis above regarding the first statements of the proof is correct and/or critique my analysis pointing out any errors or shortcomings ... ...
Peter
I am currently focussed on Section 9.7 Exact Sequences.
On page 391 A&A state and prove the Short Five Lemma. I need help with some of the details of the 'diagram chasing' in the proof.
The Short Five Lemma and the first part of its proof read as follows:https://www.physicsforums.com/attachments/3628
https://www.physicsforums.com/attachments/3629In the first two lines of the above proof we find the following:" ... Suppose
Now
Now I need someone to critique my detailed reasoning reasoning concerning these statements - I think I understand ... but then, I am working by myself on this material ... so a confirmation that I am on the right track would be most helpful ...Now ... my reasoning is as follows:
We suppose that
We need to show that
... which implies that
... which implies that
... ... and then the proof of (i) continues ...
Can someone please confirm that the details of my analysis above regarding the first statements of the proof is correct and/or critique my analysis pointing out any errors or shortcomings ... ...
Peter
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