If $(ab)^k = a^k b^k$ for two consecutives integers $k$, is $G$ abelian?
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 $n$
102154
If $a$ and $b$ are consecutive integers, prove that $a^2 + b^2 + a^2b^2$ is a perfect square.
105180
When does $(ab)^n = a^n b^n$ imply a group is abelian?
11003
Proving that $G/N$ is an abelian group
135039
${\rm Aut}(G)$ is cyclic $\implies G$ is abelian
69279
If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
70619
Supposed $a,b \in \mathbb{Z}$. If $ab$ is odd, then $a^{2} + b^{2}$ is even.
126358
Prove for $\gcd$ of two integers $a,b$, that $(a^n,b^n) = (a,b)^n$
102777
Is $G^{(X, Y)} = (G^X)^Y?$ ($A^B$ just means that $B$ is mapped to $A$)
5297
If G is a finite abelian group and $a_1,...,a_n$ are all its elements, show that $x=a_1a_2a_3...a_n$must satisfy $x^2=e$.
58654
If $|A| > \frac{|G|}{2} $ then $AA = G $
117815
Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers.
1364
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
125559
In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
20679
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
25573
If $\gcd(a,b)=1$, then $\gcd(a+b,a^2 -ab+b^2)=1$ or $3$.