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:
:the cyclic group of order $4$ 
:the Klein $4$-group
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.
== Sources ==
* : Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.8 \ \text {(b)}$
Category:Examples of Algebra Loops
Category:Examples of Cayley Tables
Category:Groups of Order 4