Does the first Sylow theorem imply groups of order p^k are cyclic?

[Silviu]

Hello! I am a bit confused about the first Sylow theorem. So it says that if you have a group of order $p^mn$, with gcd(n,p)=1, you must have a subgroup H of G of order $p^m$. So, if I have a group G of order $p^k$, there is only one subgroup of G of order $p^k$ which is G itself. Does this means that any group of order $p^k$, with p prime is cyclic?

Thank you!

 

[fresh_42]

Which other finite Abelian groups do you know which are not cyclic?