If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.
105180
When does $(ab)^n = a^n b^n$ imply a group is abelian?
11003
Proving that $G/N$ is an abelian group
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
52357
If $a,b \in G$, show that $ab^{n}a^{-1}=(aba^{-1})^{n}$.
85011
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
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?
13631
Showing group with $p^2$ elements is Abelian
108425
Problem on abelian group
63136
Prove $G$ is Abelian if $N$ is in the centre of $G$ and $G/N$ is cyclic
48393
Abelian groups of order n.
33233
Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$.
156429
If $G$ is a group whereby $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show $G$ is abelian.
48081
Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
90332
Prove that that $U(n)$ is an abelian group.
40287
Prove: if $a,b\in G$ commute with probability $>5/8$, then $G$ is abelian
138406
If N and every subgroup of N is normal in G then G/N is abelian .