# An equivalent definition of a group

Exercise 15 from Hungerford: Algebra.

Let $G$ be a nonempty finite set with an associative binary operation such that for all $a,b,c\in G\,\,ab=ac \Rightarrow b=c$ and $ba=ca \Rightarrow b=c$. Then $G$ is a group. Show that this conclusion may be false if $G$ is infinite.

I've solved the first part, but I wasn't able to find a counter-example.

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After you are done solving the problem, the following reference has more: en.wikipedia.org/wiki/Cancellative_semigroup – Jonas Meyer Jan 24 '12 at 20:08

HINT: Is there a familiar (infinite) set and commutative operation in which you know that $ab=ac$ implies $b=c$? It's an example you can really count on.
@spohreis: I don't understand what it is you are trying to do/say. What "is because" what? If you are trying to characterize what properties a binary associative operation on an infinite set must satisfy in order for it to give you a group, there are plenty of characterizations. E.g., that all equations $ax=b$ and $xa=b$ have solutions. – Arturo Magidin Jan 24 '12 at 20:38