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This question already has an answer here:

Why we study the normal subgroups? What is the importance of normal subgroups in group theory? My teacher ask today that how you think that normal subgroups are important. I told that it is necessary for the factor groups but he was not satisfied with this answer.please tell me the importance of normal subgroups. Thanks

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marked as duplicate by Dietrich Burde, Aaron Maroja, Ethan Bolker, Arnaud D., muaddib Jan 29 '18 at 15:16

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    $\begingroup$ We can't. I got asked the same question in an oral exam once, and my teacher wanted to hear "normal groups are important, because we can build the factor group." There are a lot of properties of normal subgroups (I am sure you know them), and no one can guess which one your professor cares about and which one he doesn't... $\endgroup$ – Dirk Jan 29 '18 at 12:35
  • $\begingroup$ Many nice explanations are also given here. $\endgroup$ – Dietrich Burde Jan 29 '18 at 12:48
  • $\begingroup$ Perhaps they were not satisfied because they wanted you to go on to explain why factor groups are important?...(as in Sunny Rathore's answer.) $\endgroup$ – user1729 Jan 29 '18 at 13:20
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One way you can answer for this question is normal subgroups allow you to define quotient group which usually has simpler structure than the whole group. So this is like a shadow of a group $G$ on $G/N$ for normal subgroup $N$. So if you have many normal subgroups then you can take the lots of shadows and by looking at these shadows of $G$, you can deduce some information about group $G$.

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Normal subgroups are important because they are exactly the kernels of homomorphisms.

In this sense, they are useful for looking at simplified versions of the group, via quotient groups.

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