Question

Let $\mathit{G}$be a group with this property: if $\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathit{c}\mathbf{\in }\mathit{G}$and$\mathit{a}\mathit{b}\mathbf{=}\mathit{c}\mathit{a}$, then $\mathit{b}\mathbf{=}\mathit{c}$. Prove that $\mathit{G}$is abelian.

Short answer

It is proved that $G$ is abelian.

Step-by-step solution

Write the basic properties of G

If$G$ is a group and $a,b\in G$ then,

1. ${\left(ab\right)}^{-1}=b^{-1}{a}^{-1}$;
2. ${\left(a^{-1}\right)}^{-1}=a$

Show thatGis an abelian

Consider$a,b$as arbitrary elements in $G$

Assume $ab{a}^{-1}=c$where, $c\in G$.

Multiply by $a$on both sides of the equality$ab{a}^{-1}=c$then we have,

$ab=ca$

From the given conditions, we know that, $b=c$

Thus,

$ab{a}^{-1}=b$

Multiply by$a$on both sides of the equality$ab{a}^{-1}=b$then we have,

$ab=ba$

Hence, the given statement is proved.