When will a group be Abelian?

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.

If $G$ is finite of order $n$ and $n$ is an abelian number, then $G$ is abelian.

$n$ is an abelian number when $n$ is a cubefree nilpotent number, that is, if $n = p_1^{a_1} \cdots p_r^{a_r}$, then

• $a_i < 3$
• $p_i^k \not \equiv 1 \bmod{p_j}$ for all $1 \leq k \leq a_i$

(adapted from this answer)