1. ## group question

ı have some problems in writenig mathematical language so ı need some help actually ı need the answer but ı want to learn writng my teacher always talking about my writing so please help me,

Let$\displaystyle G$ is a group,$\displaystyle {x},{y}\in{G}$
if$\displaystyle xy^2=yx^3$and$\displaystyle x^3y=yx^2$then
$\displaystyle x=y=e$

2. Originally Posted by sah_mat

Let $\displaystyle G$ be a group and $\displaystyle {x},{y}\in{G}.$ prove that if $\displaystyle xy^2=yx^3$ and $\displaystyle x^3y=yx^2,$ then $\displaystyle x=y=e.$
there are different ways to solve the problem. here's one way: we have: $\displaystyle x^3=yx^2y^{-1}=(yxy^{-1})^2,$ and: $\displaystyle xy=yx^3y^{-1}=(yxy^{-1})^3.$ thus: $\displaystyle x^9=(xy)^2.$

we also have: $\displaystyle xy=x^{-2}yx^2,$ which gives us: $\displaystyle (xy)^2=x^{-2}y^2x^2.$ hence: $\displaystyle x^9=x^{-2}y^2x^2,$ and thus: $\displaystyle x^9=y^2.$ hence $\displaystyle y=x^7$ because $\displaystyle xy^2=yx^3.$ but then

$\displaystyle x^3y=yx^2$ will gives us $\displaystyle x=e,$ and hence $\displaystyle y=e.$

3. ı am glad to see your solution i will try to do again by myself,thanks again