# Math Help - group question

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

2. Originally Posted by sah_mat

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

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

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

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