Theorem 4.16

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Statement:

If \$G\$ is a group with binary operation \$*\$, and if \$a\$ and \$b\$ are any elements of \$G\$, then the linear equations
\$a * x = b\$ and \$y * a = b\$ have unique solutions.

Proof:

First we show the existence of at least one solution by just computing that \$a' * b\$ is a solution of \$a* x = b\$.
Note that:
\$a * (a' * b) = (a * a') * b\$ by associative law
\$= e * b\$ definition of \$a'\$
\$=b\$ property of \$e\$

Thus \$x = a' * b\$ is a solution of \$a * x = b\$. In a similar fashion, \$y = b * a'\$ is a solution of \$y * a = b\$.

To show uniqueness of \$y\$, we use the standard method of assuming that we have two solutions, \$y_{1}\$ and \$y_{2}\$ so that \$y_{1} * a = b\$ and \$y_{2} * a = b\$. Then \$y_{1} * a = y_{2} * a\$ and by theorem 4.15 \$y_{1} = y_{2}\$. The uniqueness of x follows similarly.

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