TITLE: About Abelian Finite Groups and Euler's Function
QUESTION [6 upvotes]: Let $G$ be a finite group or $|G|=n$ and let $(\phi(n) ,n)=1$ (where $\phi(n) $ is Euler's function). Now prove $G$ is abelian.

REPLY [3 votes]: Every group $G$ of order $n$ such that $(n,\phi(n))=1$ is in fact cyclic, hence abelian. For an elementary proof see one of the following references (the first proof is simple enough to be suitable for an elemenatry class in group theory, the author says):
Jungnickel, Dieter. On the Uniqueness of the Cyclic Group of Order $n$. Amer. Math. Monthly, Vol. 99, No. 6 (1992)
Gallian, J. A. Moulton, David. When is $\mathbb{Z}_n$ the only group of order $n$?, Elemente der Mathematik, Vol. 48 (1993).