Group/Non-Group Examples

Examples of Algebraic Structures which are not Groups

Arbitrary Example: Order 4

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.