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September 12th, 2017, 11:07 AM   #1
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Abilization of a group

Let $G$ be a group, $N$ a normal subgroup and $(A,\rho)$ an abilization of $G$. Let $B=\rho(N)$ and $\lambda:G/N\to A/B$ be given by $\lambda(gN)=\rho(g)B$.


I want to show that $(A/B,\lambda)$ is an abilization of $G/N$ (in this problem, abilization is meant in the sense of universal property).

Thanks in advance for your help.

Last edited by skipjack; September 29th, 2017 at 08:52 AM.
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September 12th, 2017, 11:18 AM   #2
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What progress have you made on the problem? Any thoughts? Anything you've tried?

You will especially get these comments on Stackexchange. https://math.stackexchange.com/quest...ion-of-a-group

Over there you have to show that you've made some effort of your own.

Last edited by Maschke; September 12th, 2017 at 11:23 AM.
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September 12th, 2017, 11:32 AM   #3
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I am blocked from the beginning and I don't know what to do, since tomorrow I have a quiz.
please help me if you can.

Last edited by skipjack; September 12th, 2017 at 11:48 AM.
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September 12th, 2017, 12:35 PM   #4
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General steps for getting started on a proof:

* Write down the exact definition of every technical term, especially the ones that are a little fuzzy. For example I'm sure you know what a group is, but perhaps you should write down exactly what's an Abelianization, what's a normal subgroup, what's a universal property.

* Write down exactly what you need to show in order to complete the proof. In other words write down the thing that if you show it, you're done.

More often than not, in classroom situations the above two steps are enough to make the proof pretty much write itself.
Thanks from Joppy

Last edited by Maschke; September 12th, 2017 at 12:43 PM.
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September 29th, 2017, 06:08 AM   #5
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By the way, I believe the word you want is "abelianization", making an arbitrary group into an "Abelian" group, named for Niels Abel, also known as a "commutative" group, not "abilization".

Last edited by skipjack; September 29th, 2017 at 08:50 AM.
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