I What is the importance of p-Sylow subgroups in understanding group structure?

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Hi, I'm reading about p-Sylow subgroups from "Algebra" by Serge Lang and for me the definition of p-Sylow subgroups is a very specific type of subgroup, i know that find a p-Sylow subgroup isn't so weird but, what is the use of this kind of groups?
 
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If you have a finite group, then the first question that comes up is: what is the group structure? This means, which are the normal subgroups, does it split into a direct product, or at least a semidirect product, and so on.

If you look into (last attachment)
Problems with Solutions (complete).pdf on
https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/
and search for 'Sylow' (38 occurrences), then you find a lot of examples.

It is basically all about the structure of groups.
 
fresh_42 said:
If you have a finite group, then the first question that comes up is: what is the group structure? This means, which are the normal subgroups, does it split into a direct product, or at least a semidirect product, and so on.

If you look into (last attachment)
Problems with Solutions (complete).pdf on
https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/
and search for 'Sylow' (38 occurrences), then you find a lot of examples.

It is basically all about the structure of groups.
Thanks!
 

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