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Thursday, November 4, 2010

Quaternion group as a Freegroup

The Quaternion group can be defined as follows $\{(a,b) : a^4=e,b^4=e,a^2=b^2,ab=ba^3\}$. Let's be practical and set a=i, b=j and let's implement this in Mathematica.

The following Mathematica code

$r:=\{ \text{iiii} \to \text{""} , \text{jjjj} \to \text{""} ,\text{ii} \to \text{jj},\text{ij} \to \text{jiii}\}$





Let me explain. The variable $r$ is a list which contains four ( production ) rules, i.e.:
$\text{iiii} \to \text{""}$ - Says that $i^4 = 1$.
$\text{jjjj} \to \text{""}$ - Says that $j^4 = 1$.
$\text{ii} \to \text{jj}$ - Says that $i^2 = j^2$. And $i^2=-1$ as we know.
$\text{ij} \to \text{jiii}$ - Says that $ij = jiii$. Or $ij=-ji$.

The command $f[\text{x_}]:=\text{StringReplace}[x,r]$ takes a string as input and applies the production rules once from left to right. This command can be repeated until the input string no longer changes. In the case of $"ijij"$ it took $5$ times and the input and resp. outputs were as follows.


As you see $(ij)^2=k^2$ is correctly evaluated to $j^2=-1$.

From here on it's fairly easy to generate all elements from the quaternion group by string concatenation and applying the production rules.

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