Finding a suitable binary operation which satisfies all the stated properties

Question

In my abstract algebra study material/textbook I found the following question:

Let $S=\{a,b,c\}$. Define a suitable binary operation (if possible) $f:S \times S \to S$ such that it satisfies the following properties:

• $f$ is associative
• Every element in $S$ has an identity w.r.t $f$ and that identity is unique for all elements of $S$.
• Every element in $S$ has an inverse w.r.t $f$ and that inverse is unique for that element in $S$.
• $f$ is commutative on $S$ Also, show an example of $f$ if $S=\{0,1,2\}$.

I thought of binary operations like addition, subtraction, multiplication, division, but none of them seem to satisfy all the four conditions.

Next, I thought of defining the operation $f$ on $S$ as follows:

$$f(a,b)=f(a,b)=c;f(b,c)=f(c,b)=a;f(a,c)=f(c,a)=b;f(a,a)=a;f(b,b)=b;f(c,c)=c$$

But even this does not work as $f(a,f(b,c))=a$ while $f(f(a,b),c)=c$. Hence, $f$ is not associative in this case.

I have tried to come up with other examples but none of them satisfied all the stated properties.

So, my query is: Does such an $f$ even exist for a general $\{a,b,c\}$ or at least for $\{0,1,2\}$ ?

Answer 1

You treat a as 0, b as 1 and c as 2 and define the binary operation on {0,1,2} as addition module 3, under this operation it forms a cyclic group of order 3.

Answer 2

like if x*y = (x + y)modulo 3, and for general case, use the bijection between { 0,1,2} and { a,b,c}