Classification of abelianness-forcing numbers

Name
This result is attributed to Dickson, and is hence also called Dickson's theorem, though there are many results with that name.

Statement
The following are equivalent for a natural number $$n$$:


 * 1) Any group of order $$n$$ is an abelian group.
 * 2) $$n$$ has prime factorization of the form $$n = p_1^{k_1}p_2^{k_2} \dots p_r^{k_r}$$ with $$k_i \le 2$$ for all $$i$$ AND $$p_i$$ does not divide $$p_j^{k_j} - 1$$ for any $$1 \le i,j \le r$$.

Facts used

 * 1) uses::Finite non-abelian and every proper subgroup is abelian implies not simple