Sem 1 - Homework

Ex 7. Let $(G,\cdot )$ be a group. Show that: i. $G$ is abelian $\iff \forall x,y\in G,(x,y{)}^2=x^2{y}^2$

$$(xy{)}^2=x^2{y}^2\iff xyxy=x^2{y}^2\underset{|\cdot y^{-1}}{\iff }xyx=x^{2}y\underset{|\cdot x^{-1}}{\iff }yx=xy\iff Gabelian$$

ii. If $x^2=1$ for every $x\in G$, then $G$ is abelian.

\text{Let } x,y\in G\implies xy\in G\implies (xy)^2=1\ (1)\\ x^2=1, y^2=1\implies x^2y^2=1\ (2)\\ (1),(2)\implies (xy)^2=x^2y^2\overset{i.}{\implies}G \text{ abelian} \end{align}