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

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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

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