
Algebra question
Hi guys,
I am working on an algebra problem and am not quite sure if this is right. It "feels" correct but at the same time I kind of feeling like I might be begging the question.
$\displaystyle \text{\underline{Theorem:} Let G be a group with the following property:}$ $\displaystyle \text{Whenever
a, b, c}$ $\displaystyle \in$ $\displaystyle \text{G and a*b = c*a implies b = c, then G is Abelian}$
$\displaystyle \text{\underline{Proof:} Suppose a*b = c for some a, b, c } $ $\displaystyle \in$ $\displaystyle \text{ G.Applying a both sides by a on the right gives a*b*a = c*a.}$
$\displaystyle \text{The associative property of groups gives a*(b*a) = c*a.
By hypothesis,}$
$\displaystyle \text{we can cancel a from both sides
yielding b*a = c.
Hence, b*a = c}$$\displaystyle \text{and a*b = c
implies a*b = b*a. Therefore, G is Abelian.}$ $\displaystyle \blacksquare$
