Show that G is a group, if G is finite, the operation is associative, and cancellation law holds

Let $G$ be a non-empty finite set with an associative binary operation so that cancellation law holds, i.e. $ab=ac$ or $ba=ca$ implies $b=c$, for any choices of $a,b,c$ in $G$. Assume that there is an identity element $e$ in $G$. Show that $G$ is a group.

Proof: To show $G$ is a group, conditions must hold. Suppose that $ab=ac$, then $b=eb=(a^{-1}a)b =a^{-1}(ab)=a^{-1}(ac) =(a^{-1}a)c=ec=c$. So left cancellation holds in $G$.

2) Suppose $ba=ca$, then $b=be=b(aa^{-1}) =(ba)a^{-1}=(ca)a^{-1} =c(aa^{-1})=ce=c$. So right cancellation holds in $G$.

3) If $a$ exists in $G$ then $aa^{-1}=a^{-1}a=e$, where $a$ is an inverse of $a^{-1}$. Since inverses are unique, $(a^{-1})^{-1}=a$.

4) Let $x$ be the inverse of $ab$. Then $(ab)x=e$. By associativity we have $a(bx)=aa^{-1}$. Through left cancellation we have $bx=a^{-1}bx=ea^{-1}=b(b^{-1}a^{-1})$ and $x=b^{-1}a^{-1}$.

Thus $(ab)^{-1}=b^{-1}a^{-1}$. So all conditions hold, $G$ is a group.

Is this proof correct? I know to show $G$ is a group these conditions have to be met. This is all I have to show right?

Solution 1:

Another proof makes use of the pigeonhole principle. Fix $a \in G$, and consider the map $$ f : G \to G, \qquad x \mapsto a x. $$ Since $G$ is cancellative, the map is injective, hence surjective because $G$ is finite, so that there is $b \in G$ such that $e = f(b) = a b$.

Similarly, there is $c \in G$ such that $c a = e$.

Thus $$c = c e = c (a b) = (c a) b = e b = b,$$ and $b = c$ is the required inverse of $a$.

You need not assume that there is an identity, but can prove that one exists.

First show, with the arguments above, that any element $z$ of $G$ can be written in the form $z = x a$, for some $x \in G$. Then show that there is $s \in G$ such that $a s = a$. Now $$z s = x a s = x a = z, \quad\text{for all $z \in G$,}$$ so $s$ is a right identity. Similarly there is a left identity $t$ such that $$t z = z,\quad \text{for all $z \in G$,}$$ and finally $$t = t s = s$$ is the identity.

Solution 2:

You know that cancellation holds in $G$, it's a hypothesis, so you have not to prove it. And, most important, you can't use inverses of elements without first proving they exist, which is exactly what you have to do.

Consider an element $a\in G$ and the map $f_a\colon G\to G$ defined by $$ f(x)=ax $$ This map is injective. Why?

This map is also surjective, because of an important hypothesis you have. Which one?

Now, what can you say about the element $x$ such that $f_{a}(x)=e$? It is…

Next, consider the map $g_{a}\colon G\to G$ defined by $g_{a}(x)=xa$. Repeat the reasoning to conclude.

Solution 3:

No, you seem to be assuming things like the inverse would exist in your proof, which is the only thing you really need to prove (the other 3 conditions in the definition of a group are given to you).

So choose $a\in G$, and consider the set $$ \{a, a^2, a^3, \ldots \} \subset G $$ Since $G$ is finite, this set is finite, and so there exist $a^i, a^j$ such that $i>j$ and $$ a^i = a^j $$ Now consider $f = a^{i-j}$, then $$ fa^j = a^i = a^j = ea^j \Rightarrow f = e $$ Hence, if $b = a^{i-j-1}$, then $$ ab = e = ba $$ and hence $b = a^{-1} \in G$