Group/Non-Group Examples/Arbitrary Order 4
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Example of Algebraic Structure which is not a Group
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.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{B vi}$