I've been reading about Number Theory and I came across this proof that the finite subgroups of the multiplicative group of a field is cyclic. However, it seems the proof applies to all finite groups so please tell me where I go wrong.
Let $G$ be a generic group of order $n$. By Lagrange the order of the elements of the group must divide $n$. Let $\psi(d)$ count the number of elements in the group of order $d$. Since $\sum_{d|n} \psi(d)=n$ the Möbius Inversion theorem tells us that this function is just Euler's totient function. Since this function is always one or greater, we always have that there exists an element of order exactly $n$, so the group is cyclic.
Obviously I know the answer is no, but I thought using this title would make things more interesting.