Prove that if $G$ is a group with $(ab)^{2}=a^{2}b^{2}$ for all $a,b\in G$, then $G$ is abelian.
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
83938
If$(ab)^n=a^nb^n$ & $(|G|, n(n-1))=1$ then $G$ is abelian
139100
A question on the group algebra $k[G]$ of a finite abelian group
33593
An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$
57421
If $\forall x \in G$ where G is a group $x=x^{-1}$ then how is this group abelian?
127603
If $G$ be a group of order $8$ and $x$ be a element of order $4$. Prove that $x^2 \in Z(G)$
79933
Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
76317
Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian
74586
Prove that every group of order $4$ is abelian
82037
Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?
60208
Proof of property of abelian group
80601
Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
68971
Elementary approach to proving that a group of order 9 is Abelian
101543
Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.
23394
If $G/Z(G)$ is cyclic, then $G$ is abelian
94239
Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
52357
If $a,b \in G$, show that $ab^{n}a^{-1}=(aba^{-1})^{n}$.