Klein Four-Group as Subgroup of S4

Theorem

Let $G$ be the following subset of the symmetric group on $4$ letters $S_4$, expressed in two-row notation:

$\ds e$

$=$

$\ds \begin{bmatrix} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{bmatrix}$

$\ds a$

$=$

$\ds \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{bmatrix}$

$\ds b$

$=$

$\ds \begin{bmatrix} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{bmatrix}$

$\ds c$

$=$

$\ds \begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{bmatrix}$

Then $G$ is an example of the Klein $4$-group.

By inspection, the Cayley table is constructed:

$\begin{array}{c|cccc}

 & e & a & b & c \\

\hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & e & a \\ c & c & b & a & e \\ \end{array}$

Again by inspection this can be seen to be the same as the Cayley table for the Klein $4$-group.

$\blacksquare$

1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 44$. Some consequences of Lagrange's Theorem: Illustration $2$