TITLE: When will a group be Abelian?
QUESTION [16 upvotes]: 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?

REPLY [4 votes]: (1) A group G is abelian iff G is equal to its center
(2) A group G is abelian iff G is isomorphic to an abelian group.
(3) If G is a group and every element in G has order 1 then G is abelian.
(4) If G is a group and every element in G has order 2 then G is abelian.
(5) Every cyclic group is abelian.
(6) If a group G has prime order then it is abelian.
Proof of (6): Let G be a group of prime order, p. Let $g\in G$ be such that $g$ is not the identity, by lagranges theorem , $<g>$ divides $|G|$. Since the order of G is prime, the order of $<g>$ is either 1 or p. Since $g$ is not the the identity, it follows that G is cyclic, and thus abelian by (5).
(7) A subgroup H of G is abelian if G is abelian. 
(8) Let G be a finite group and H an abelian subgroup. If the order of H is the order of G then G is abelian.