Question

If $\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathit{c}\mathbf{\in }\mathit{G}$, prove that there is a unique element$\mathit{x}\mathbf{\in }\mathit{G}$ such that $\mathit{a}\mathit{x}\mathit{b}\mathbf{=}\mathit{c}$.

Short answer

It is proved that there is a unique element $x\in G$.

Step-by-step solution

Write the properties of G

If $G$ is a group and $a\in G$ an element of finite order$n$then, $a^k=e$ if and only if $n\left|k\right.$.

Show that there is a unique elementx∈G such that axb=c

Consider$axb=c$.

Multiply by $a^{-1}$on the left side and by $b^{-1}$on the right side of the given equation. Then, we get,

$x=a^{-1}c{b}^{-1}$

Thus, if there exist $x$ such that it satisfies $axb=c$then, $x=a^{-1}c{b}^{-1}$.

Hence, the given statement is proved.