A Cayley table is an operation table for a group. The entry \(ab\) in the row labeled with \(a\) and the column labeled with \(b\) is the result of applying first \(b\) and then \(a\text{.}\)
\begin{equation*}
\begin{array}{c|cccccccc}
\amp e \amp r \amp r^2 \amp r^3 \amp h \amp v \amp d \amp d' \\
\hline
e \amp e \amp r \amp r^2 \amp r^3 \amp h \amp v \amp d \amp d' \\
r \amp r \amp r^2 \amp r^3 \amp e \amp d' \amp d \amp h \amp v \\
r^2 \amp r^2 \amp r^3 \amp e \amp r \amp v \amp h \amp d' \amp d \\
r^3 \amp r^3 \amp e \amp r \amp r^2 \amp d \amp d' \amp v \amp h \\
h \amp h \amp d \amp v \amp d' \amp e \amp r^2 \amp r \amp r^3 \\
v \amp v \amp d' \amp h \amp d \amp r^2 \amp e \amp r^3 \amp r \\
d \amp d \amp v \amp d' \amp h \amp r^3 \amp r \amp e \amp r^2 \\
d' \amp d' \amp h \amp d \amp v \amp r \amp r^3 \amp r^2 \amp e \\
\end{array}
\end{equation*}
To find \(rdv\text{,}\) we choose a grouping: \((r d) v\text{.}\) Then \(rd\) is the entry in row \(r\) and column \(d\text{,}\) which is \(h\text{.}\) Then we find \(hv\text{,}\) which is the entry in row \(h\) and column \(v\text{,}\) resulting in \(r^2\text{.}\)
Let \(G\) be a group. The order of \(G\) is \(|G|\text{,}\) its cardinality. The order of an element \(g\in G\) is the smallest integer \(n>0\) such that \(g^n=e\) if such an integer exists or \(\infty\) otherwise and is denoted \(|g|\text{.}\)
We have \(|D_4|=8\) since there are eight elements. The identity \(e\) has order one. All four reflections \(h,v,d\text{,}\) and \(d'\) have order two, as does the 180° rotation \(r^2\text{.}\) The remaining two rotations \(r\) and \(r^3\) have order four.
\((\Z,+)\) is an infinite group. \(0\) has order 1, but all other integers have infinite order because no finite sum of a non-zero integer will ever be zero.
