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properties of subgroups

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Let A be a group and let Z be a subgroup of A. Let x,y be elements of G. Tell if each statement is true of false and give reason

1) if Zx = Zy then xZ = yZ
2) if Zx = Zy then Zxy^-1 = Z
3) if Zx = Zy then Zx^2=Zy^2

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Properties of subgroups are carefully evaluated in this solution.

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Zx is the set of all g in G that can be expressed as zx for some z in Z. Then, we have: Zx = Zy means that for every g=z_1x there's z_2 in Z such that g = z_2y. In other words, z_1 x = z_2 y, xy^{-1} = (z_1^{-1})z_2, and thus xy^{-1} is in Z.

1. False. The simplest counterexample I was able to find is the group S_3 of ...

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