a, b are in G, a group. H is a subgroup. If aH = bH then Ha^(-1) = Hb^(-1). Prove/disprove. I can't even tell if this is true or false so I really can't prove it!
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Originally Posted by fulltwist8 a, b are in G, a group. H is a subgroup. If aH = bH then Ha^(-1) = Hb^(-1). Prove/disprove. I can't even tell if this is true or false so I really can't prove it! Hint: $\displaystyle xH = yH$ if and only if $\displaystyle x^{-1}y \in H$ and same situation with $\displaystyle Hx=Hy$.
Last edited by ThePerfectHacker; Feb 27th 2008 at 06:25 PM.
Originally Posted by ThePerfectHacker Hint: $\displaystyle xH = yH$ if and only if $\displaystyle xy^{-1} \in H$ $\displaystyle xH = yH\ \Leftrightarrow\ \color{red}y^{-1}x\color{black}\in H$
Originally Posted by JaneBennet $\displaystyle xH = yH\ \Leftrightarrow\ \color{red}y^{-1}x\color{black}\in H$ It makes no difference. Because if $\displaystyle y^{-1} x\in H$ then $\displaystyle (y^{-1} x)^{-1} = x^{-1}y\in H$. EDIT: Okay, I see I wrote it the other way around.
ok thanks, i figured that out!!... but what about if it's "If aH = bH, then a^2 H = b^2 H"?
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