Group/Non-Group Examples/Arbitrary Order 4

Example of Algebraic Structure which is not a Groups
Let $S = \set {1, 2, 3, 4}$.

Consider the algebraic structure $\struct {S, \circ}$ given by the Cayley table:


 * $\begin{array}{r|rrrr}

\circ & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ 2 & 2 & 4 & 3 & 1 \\ 3 & 3 & 2 & 4 & 3 \\ 4 & 4 & 3 & 1 & 2 \\ \end{array}$

Then $\struct {S, \circ}$ is not a group.

Proof
It is immediately seen that $\struct {S, \circ}$ violates the Latin Square Property.

For example, both the column headed $3$ and the row headed $3$ have $2$ instances of $3$ in them.