Constant Mapping/Examples/Constant Mappings on Set of 3

Examples of Constant Mappings

Let $X = \set {a, b, c}$.

Let $S = \set {f_a, f_b, f_c}$ be the constant mappings from $X$ to $X$.

The Cayley table for composition on $S$ is as follows:

$\begin{array}{c|cccc}

\circ & f_a & f_b & f_c \\ \hline f_a & f_a & f_a & f_a \\ f_b & f_b & f_b & f_b \\ f_c & f_c & f_c & f_c \\ \end{array}$

As can be seen, there is no identity element, so $\struct {S, \circ}$ is not a group.

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Exercise $3$