**Problem.** Let be a group of order 225. By Sylow theorem has a unique Sylow 5-subgroup Prove that if is cyclic, then is abelian.

**Solution.** Choose any Sylow 3-subgroup and let So and both and are abelian. In order to prove that is abelian, we only need to prove that for any Now for some because is normal in Also because . Thus and so Hence because It is easy to see that has only one solution, i.e. Therefore and so

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i didn’t get that part (b^9)a(b^-9)=(a)^(k^9)

I have not ideas about sylow 5- subgroup and sylow 3 -subgroup of a group of order 225.

Why b^9ab^-1=a^k?

I didn’t say that! I said and that’s because is normal.