$G=\{e,x,y\}$ is a group and $e$ is the identity element of this group. Using the operation table show that whether this group is commutative and cyclic or not.
This is an exercise question in my book. The thing I do not understand is the operation is not given in the question. So how will I make the operation table ?
Best Answer
Well, whatever this operation is, you have $ex= xe = x$ and $ey=ye=y$ so you only need to figure out what is $x^2,y^2,xy$ and $yx$?
Since order $r\ne 1$ of the element must divide the order of a group (which is $3$) we see that $x^3=e$, so subgroup $\{e,x,x^2\}$ is equal to $G$ and thus $x^2=y$ and simillary $y^2=x$.
Now what si $xy$? It is simply $e$ since $xy= x\cdot x^2 = x^3=e$.
\begin{matrix} *&e&x&y&\\ e&e&x&y&\\ x&x&y&e&\\ y&y&e&x& \end{matrix}
So this group is cyclic (and commutative).