Why is the 'Law of Cancellation' for groups only an implication?

Question

It is easy to see that for a Group $G$ and $a,b \in G$

$ab = ac \Rightarrow b = c$

(See also here)

But what is about the other direction? That is:

$b = c \Rightarrow ab = ac$

Does this implication hold as well?

Answer 1

Of course the other direction holds. The other direction has nothing to do with groups or with anything really; it's just what equality means! Saying $a=b$ means that $a$ and $b$ are exactly the same thing; if $a$ and $b$ are the same thing then $ac$ and $bc$ are the same.

Answer 2

Yes, it does. In a group, you are allowed to left-multiply the expression $b = c$ with the group element $a$ to get $ab = ac$.

Answer 3

Yes the other way holds. This is essentially just the fact that multiplication is well-defined. The reason that they do not list the cancellation law as an if and only if is that the other way doesn't actually have to do with cancellation.

Answer 4

The group multiplication is a binary operation on the set of elements of the group, associating to each pair $\langle u,v\rangle$ of elements of $G$ a new element $uv$. When you have $b=c$, you are fixing, let's say, the right argument of this binary operation, and what you are left with is an ordinary function of one argument: the left-multiplication by elements of $G$. As you certainly know, a function associates to each element of its domain ($G$ in this case) one and only one element of its image (again $G$).