Facts about Abelian Groups and group order. I look for some theorems which tell us about the relation between the property of being abelian for groups and the order of the group. 
I think these theorems are provided in a second course of group theory or more advanced courses.  At this stage, however, I'm interested in knowing some of these theorems without the details of proofs of this theorems.
I hope that you can give me some of these theorems!
Thanks.
 A: Here are some facts.  Assume all groups below are finite.


*

*All finite abelian groups look like this: $\mathbb{Z}_{p_1^{e_1}}\oplus \mathbb{Z}_{p_2^{e_2}}\oplus \mathbb{Z}_{p_3^{e_3}}\oplus \cdots \oplus \mathbb{Z}_{p_n^{e_n}}$, where the $p_i$ are not necessarily distinct.

*$\mathbb{Z}_{pq}\cong \mathbb{Z}_p\oplus\, \mathbb{Z}_q$ if and only if $p$ and $q$ are coprime.

*A group $G$ is cyclic if and only if it has exactly one subgroup of order $d$ for every divisor $d$ of $|G|$.

*Lagrange's theorem states that whenever $H$ is a subgroup of $G$, the order of $H$ divides the order of $G$.  The converse does not hold in general, but it does for abelian groups - that is, for every divisor $d$ of $|G|$, there is a subgroup of order $d$ in $G$.

*If $3/4$ths or more of a group's elements have order $2$, the group is abelian.  In particular, if every element in a group has order $2$, that group is abelian.  (Note: the latter obviously follows from the former, but is proved much more easily by itself.)

*The derived subgroup and inner automorphism group of an abelian group have order $1$.

*Of all groups of order $p^e$, groups which look like $\underbrace{\mathbb{Z}_p\oplus \mathbb{Z}_p \oplus \cdots \oplus \mathbb{Z}_p}_{\text{e times}}$ have the largest number of subgroups.  Groups which look like $\mathbb{Z}_{p^e}$ have the fewest.  (This includes nonabelian groups.)

*The order of the automorphism groups of the last bullet point are $\prod_{k=0}^{e-1}p^e-p^k$ and $p^{e-1}(p-1)$, respectively.  In general the order of the automorphism group of $\mathbb{Z}_n$ is $\varphi(n)$.

*If a group is simple and solvable, it is cyclic of prime order.
A: One can classify those integers $n$ for which every group of order $n$ is abelian, see here, the last answer by Robin Chapman. This has also appeared here and there on math.stackexchange.
