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Solve $abxcd = y$ for $x$ in a group.

I just used the inverses of a,b,c,d and multiplying on both sides to get $x = b^{-1}a^{-1}yd^{-1}c^{-1}$

is this correct , or close to it? If not, whats wrong and how is it fixed.

Seth
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Achilles
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1 Answers1

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Yes, it's fine. One way to check this, using associativity and the fact that the multiplying by the identity doesn't change anything, is to compute \begin{align*} ab(b^{-1} a^{-1} y d^{-1} c^{-1}) cd &= abb^{-1} a^{-1} y d^{-1} c^{-1} cd \\ &= aa^{-1} y d^{-1} d \\ &= y \end{align*}