Multiplicative Group of Reduced Residues Modulo 8 is Klein Four-Group

Theorem

Let $K_4$ denote the Klein $4$-group.

Let $R_4$ be the multiplicative group of reduced residues Modulo $8$.

Then $K_4$ and $R_4$ are isomorphic algebraic structures.

Proof

Establish the mapping $\phi: K_4 \to R_4$ as follows:

\(\ds \map \phi e\)

\(=\)

\(\ds \eqclass 1 8\)

\(\ds \map \phi a\)

\(=\)

\(\ds \eqclass 3 8\)

\(\ds \map \phi b\)

\(=\)

\(\ds \eqclass 5 8\)

\(\ds \map \phi c\)

\(=\)

\(\ds \eqclass 7 8\)

From Isomorphism by Cayley Table, the two Cayley tables can be compared by eye to ascertain that $\phi$ is an isomorphism:

Cayley Table of Klein $4$-Group

The Cayley table for $K_4$ is as follows:

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

Multiplicative Group of Reduced Residues Modulo $8$

The Cayley table for $S$ is as follows:

$\begin{array}{r|rrrr} \times_8 & \eqclass 1 8 & \eqclass 3 8 & \eqclass 5 8 & \eqclass 7 8 \\ \hline \eqclass 1 8 & \eqclass 1 8 & \eqclass 3 8 & \eqclass 5 8 & \eqclass 7 8 \\ \eqclass 3 8 & \eqclass 3 8 & \eqclass 1 8 & \eqclass 7 8 & \eqclass 5 8 \\ \eqclass 5 8 & \eqclass 5 8 & \eqclass 7 8 & \eqclass 1 8 & \eqclass 3 8 \\ \eqclass 7 8 & \eqclass 7 8 & \eqclass 5 8 & \eqclass 3 8 & \eqclass 1 8 \\ \end{array}$

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 7$: Isomorphic Groups: Example $2$