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Math Help - 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 G is a group, {x},{y}\in{G}
    if xy^2=yx^3and x^3y=yx^2then
    x=y=e
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  2. #2
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    Quote Originally Posted by sah_mat View Post

    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.
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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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