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Thread: group question

  1. #1
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    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
    $\displaystyle x=y=e$
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  2. #2
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    Quote Originally Posted by sah_mat View Post

    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 $\displaystyle y=e.$
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  3. #3
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    ı am glad to see your solution i will try to do again by myself,thanks again
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