If $G$ is a group in which $(ab)^i = a^ib^i$ for two consecutive integers $i$, for all $a,b \in G$, can we conclude that $G$ is abelian?
117815
Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers.
21217
$G$, group. $a,b\in G$. Show that $|a|=|a^{-1}|; |ab|=|ba|$, and $|a|=|cac^{-1}|, \forall c\in G$.
126893
A group of order 2p (p prime) and other conditions - prove abelian.
33233
Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$.
145127
For any integers $m,n>1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?
90332
Prove that that $U(n)$ is an abelian group.
139934
In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
13631
Showing group with $p^2$ elements is Abelian
65900
Show that a finite group with certain automorphism is abelian
57809
For which $n$, $G$ is abelian?
26624
Prove that is group is abelian
108425
Problem on abelian group
48585
Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
66313
$\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$ if $(a,b)=1$
134403
Let $G$ be a group, and let $n\in\mathbb Z$. Show that $(ab)^n=a^nb^n\iff(ab)^{1-n}=a^{1-n}b^{1-n}$.
74586
Prove that every group of order $4$ is abelian