Methods to show a group is 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?

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