Algebra Loop/Examples/Order 4
Example of Algebra Loop
The following are the Cayley tables of the operations $\circ$ on $S = \set {e, a, b, c}$ such that $\struct {S, \circ}$ is an algebra loop whose identity is $e$:
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & b & c & e \\ b & b & c & e & a \\ c & c & e & a & b \\ \end{array} \qquad \begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & e & a \\ c & c & b & a & e \\ \end{array} \qquad \begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & c & e & b \\ b & b & e & c & a \\ c & c & b & a & e \\ \end{array} \qquad \begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & a & e \\ c & c & b & e & a \\ \end{array}$
The first two of these are the Cayley tables of:
while the $3$rd and $4$th are also isomorphic to the cyclic group of order $4$.
Proof
The initial specification allows us to populate the first few elements of the Cayley table:
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & & & \\ b & b & & & \\ c & c & & & \\ \end{array}$
Let us consider $a \circ a$.
This cannot be $a$ as there is already an $a$ in the row and column.
- Let $a \circ a = e$
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & & \\ b & b & & & \\ c & c & & & \\ \end{array}$
This immediately forces:
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & & \\ c & c & b & & \\ \end{array}$
There are two ways to complete this.
Either $b \circ b = e$ which gives us the Klein $4$-group:
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & e & a \\ c & c & b & a & e \\ \end{array}$
or $b \circ b = a$, which gives us:
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & a & e \\ c & c & b & e & a \\ \end{array}$
- Let $a \circ a = b$
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & b & & \\ b & b & & & \\ c & c & & & \\ \end{array}$
We note that $a \circ b = e$ will not work, because that forces $a \circ c = c$ which is not allowed because there is already a $c$ in the $c$ column.
Hence we have $a \circ b = c$, which forces the completion of the cyclic group of order $4$:
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & b & c & e \\ b & b & c & e & a \\ c & c & e & a & b \\ \end{array}$
- Let $a \circ a = c$
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & c & & \\ b & b & & & \\ c & c & & & \\ \end{array}$
This forces the completion of the following:
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & c & e & b \\ b & b & e & c & a \\ c & c & b & a & e \\ \end{array}$
We then note that we can rearrange the order of the rows and columns of the remaining two tables to demonstrate that they are the cyclic group of order $4$:
Let us take:
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & c & e & b \\ b & b & e & c & a \\ c & c & b & a & e \\ \end{array}$
Rearranging the order of the rows and columns as follows:
- $\begin{array}{r|rrr} \circ & e & b & c & a \\ \hline e & e & b & c & a \\ b & b & c & a & e \\ c & c & a & e & b \\ a & a & e & b & c \\ \end{array}$
from which the cyclic group of order $4$ is verified by inspection.
Similarly:
- $\begin{array}{r|rrr} \circ & e & a & b & c\\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & a & e \\ c & c & b & e & a \\ \end{array}$
Rearranging the order of the rows and columns as follows:
- $\begin{array}{r|rrr} \circ & e & b & a & c \\ \hline e & e & b & a & c \\ b & b & a & c & e \\ a & a & c & e & b \\ c & c & e & b & a \\ \end{array}$
from which the cyclic group of order $4$ is also verified by inspection.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.8 \ \text {(b)}$