Let $G$ be a group such that for all $x,y \in G, (xy)^3 = x^3y^3$ and there's no element with order $3$. Show that $xy^2=y^2x$ and $G$ is abelian.
11113
Group With an Endomorphism That is "Almost" Abelian is Abelian.
31380
Exponent and abelian groups
127603
If $G$ be a group of order $8$ and $x$ be a element of order $4$. Prove that $x^2 \in Z(G)$
46342
Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism
41977
If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$
7015
Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian
156319
Show that a group with $|G| = 33$ contains an element of order $3$
60075
Does there exist an $n$ such that all groups of order $n$ are Abelian?
1577
Order of nontrivial elements is 2 implies Abelian group
68971
Elementary approach to proving that a group of order 9 is Abelian
43580
Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G,G]$.
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
71420
Prove that a group is abelian.
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
3907
Abelian quotient group
26148
A group such that $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$ ($m$, $n$ coprime) is abelian?