Chinese reminder theorem Vs Sylow Bonjour everyone!
I wanna know the difference between Chinese Remainder Theorem and p-Sylow's subgroup "factorization". Is there a way to use one for the proof of the other. Thanks.

I'm guessing the factorization using Sylow's theorems is the one that if $G$ has a normal sylow for all primes dividing the order of $G$, then $G$ is a direct sum of its (unique) normal Sylows.
This bears little resemblance to the CRT for the following reasons:


*

*It's for finite groups, but the CRT applies to rings, even infinite ones.

*It means the factors will always be of prime order, which is not necessarily the case for the CRT


I guess you may have been fooled into thinking there was a connection since it seems to work well for $\mathbb Z_n$. There it is true that the Sylow factorization as a group amounts to the same thing as viewing it as a ring and splitting it up along its prime-power components. But this is a rather narrow, nice case that doesn't generalize.
