Intereting Posts

Characterization of lim sup, lim inf
Square Root Inequality
Matlab Recursion Loop
Is a field determined by its family of general linear groups?
Where can I learn “everything” about strange attractors?
Open mapping theorem and second category
Explanation and Proof of the fourth order Runge-Kutta method
What is the difference between outer measure and Lebesgue measure?
Is there a nonnormal operator with spectrum strictly continuous?
How to find a quadratic form that represents a prime?
Equivalent Topologies
Maximum amount willing to gamble given utility function $U(W)=\ln(W)$ and $W=1000000$ in the game referred to in St. Petersberg's Paradox?
The area of the region $|x-ay| \le c$ for $0 \le x \le 1$ and $0 \le y \le 1$
Proof that $26$ is the one and only number between square and cube
Adjoint functors preserve (co)products

According to Wikipedia, a *cyclic number* (in group theory) is one which is coprime to its Euler phi function and is the necessary and sufficient condition for any group of that order to be cyclic. Why is that true?

I can see that if $n$ is prime, that guarantees any group of order $n$ is cyclic, but I don’t seem to see how to extend it to $(n,\phi(n))=1$

It would be nice if someone could explain it to me. Thanks.

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- Primes of the form $x^2+ny^2$
- $H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.

Suppose $\gcd(n, \phi(n)) > 1$. If $n$ is not squarefree, there exists a prime $p$ such that $p^2$ divides $n$. Then $H = \mathbb{Z}_p \times \mathbb{Z}_p$ is not cyclic and neither is $H \times \mathbb{Z}_{\frac{n}{p^2}}$. If $n$ is squarefree, there exist prime divisors $p$ and $q$ of $n$ such that $q$ divides $p-1$. Then there exists a non-abelian group $H$ of order $pq$ and $H \times \mathbb{Z}_{\frac{n}{pq}}$ is not cyclic.

The other direction is not so easy to answer, but I’ll give you a few good references.

Jungnickel and Gallian give elementary proofs (or at least rough outlines for a proof) in these two papers:

Jungnickel, Dieter. On the Uniqueness of the Cyclic Group of Order $n$. Amer. Math. Monthly, Vol. 99, No. 6 (1992) JSTOR

Gallian, J. A. Moulton, David. When is $\mathbb{Z}_n$ the only group of order $n$?, Elemente der Mathematik, Vol. 48 (1993) Link to article

Pakianathan and Shankar have a paper that goes beyond cyclic numbers and gives a number theoretic characterization for abelian, nilpotent and solvable numbers too:

Pakianathan, Jonathan. Shankar, Krishnan. Nilpotent Numbers. Amer. Math. Monthly, Vol. 107, No. 7 (2000) JSTOR

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