groups

Let (G ,*) be a group and for any x, y $ (xy)^3=x^3y^3 $ , $ (xy)^4=x^4y^4 $ and $ (xy)^5=x^5y^5 $
then xy=yx.

We have from $ (xy)^5=x^5y^5 $ $ x(yx)^4y=x*x^4y^4*y $ so $ (yx)^4=x^4y^4=(xy)^4 $
In the same way from $ (xy)^4=x^4y^4 $ we have $ (yx)^3=x^3y^3=(xy)^3 $
But if a=b then $ a^{-1}=b^{-1} $ and $ (a^3)^{-1}=a^{-1}a^{-1}a^{-1}=a^{-3} $ so from $ a^3=b^3 $ we have $ a^{-3}=b^{-3} $
Then we also have $ (yx)^{-3}=(xy)^{-3} $
Now $ yx=(yx)^{4}(yx)^{-3}=(xy)^{4}(xy)^{-3}=xy $

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