When will a group be Abelian?

My Attempt:

$1.$ $G$ is abelian if and only if the mapping $g\mapsto g^{-1}$ is an isomorphism on the group $G$.

$2.$If $G$ is finite and every irreducible character is linear then $G$ is abelian.

$3.$If $\operatorname{Aut}(G)$ acts on the set $G-\{e\}$ transitively then $G$ is abelian.

$4.$If $\mathbb Z_2$ acts by automorphism on a finite group $G$ fixed point freely then $G$ is abelian.

$5.$ If $\forall a,b\in G$ $ ab=ba$ then $G$ is Abelian.

My Question:

The above are the things which I already use to show a group will be Abelian.

Is/are there any other way(s) to show a group $G$ to be Abelian?

Solution 1:

A group $G$ is abelian if and only if the multiplication map $\circ:G\times G\to G$ is a homomorphism.

If $G/Z(G)$ is cyclic, then $G$ is abelian.

and its corollary for finite groups:

If $|Z(G)| > \frac {1}{4} |G|$, then $G$ is abelian.