What is the relationship between non-repeated factors of the order of a group and its cyclicity?

Question

Consider $\mathbb F_{2^4}$ as represented by the reduction polynomial $f(z) = z^4 +z +1$. The elliptic curve $E : y^2 +xy = x^3 +z^3x^2 +(z^3 +1)$ defined over $\mathbb F_{2^4}$ has $\#E(\mathbb F_{2^4} ) = 22$. Since $22$ does not have any repeated factors, $E(\mathbb F_{2^4} )$ is cyclic.

I don't understand what is the relationship b/w non-repeated factors of the order of a group and the cyclicity of that group?

The above example is taken from the "Guide to Elliptic Curve Cryptography", page#$84$, example $3.14$.

Answer 1

I'm not too fluent in elliptic curves, but I think it is just an elementary result of group theory:

1. Any finite abelian group can be written as direct product of cyclic groups
2. the direct product of two cyclic groups $C_n\times C_m$ is isomorphic to $C_{nm}$ if and only if $gcd(n,m)=1$

Together, this means that that whenever there are no repeated factors in the order of a group, there is only one single group of that order namely the cyclic group.