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Directory:Jon Awbrey/Papers/Dynamics And Logic (view source)
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==Note 17==
==Note 17==
So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group. This is a group of six elements, say, <math>G = \{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h}, \operatorname{i}, \operatorname{j} \},\!</math> with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, <math>X = \{ a, b, c \},\!</math> usually notated as <math>G = \operatorname{Sym}(X)</math> or more abstractly and briefly, as <math>\operatorname{Sym}(3)</math> or <math>S_3.\!</math> The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in <math>\operatorname{Sym}(X).</math>
So long as we're in the neighborhood, we might as well take in
some more of the sights, for instance, the smallest example of
<br>
a non-abelian (non-commutative) group. This is a group of six
elements, say, G = {e, f, g, h, i, j}, with no relation to any
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
other employment of these six symbols being implied, of course,
|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ a, b, c \}</math>
and it can be most easily represented as the permutation group
|- style="background:#f0f0ff"
on a set of three letters, say, X = {a, b, c}, usually notated
| width="16%" | <math>\operatorname{e}</math>
as G = Sym(X) or more abstractly and briefly, as Sym(3) or S_3.
| width="16%" | <math>\operatorname{f}</math>
| width="16%" | <math>\operatorname{g}</math>
| width="16%" | <math>\operatorname{h}</math>
| width="16%" | <math>\operatorname{i}</math>
| width="16%" | <math>\operatorname{j}</math>
|-
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
a & b & c
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
c & a & b
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
b & c & a
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
a & c & b
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
c & b & a
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
b & a & c
\end{matrix}</math>
|}
<br>
Here is the operation table for S_3, given in abstract fashion:
Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
{| align="center" cellpadding="6" width="90%"
| align="center" |
<pre>
Symmetric Group S_3
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
| o |
| ^ |
| e / \ e |
| e / \ e |
| / \ |
| / \ |
| j / \ / \ / \ / \ / \ / \ j |
| j / \ / \ / \ / \ / \ / \ j |
| / \ / \ / \ / \ / \ / \ |
| / \ / \ / \ / \ / \ / \ |
| o j \ j \ j \ i \ h \ j o |
| ( j \ j \ j \ i \ h \ j ) |
| \ / \ / \ / \ / \ / \ / |
| \ / \ / \ / \ / \ / \ / |
| \ / \ / \ / \ / \ / \ / |
| \ / \ / \ / \ / \ / \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| o |
| v |
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
|}
By the way, we will meet with the symmetric group <math>S_3\!</math> again when we return to take up the study of Peirce's early paper "On a Class of Multiple Algebras" (CP 3.324–327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227–323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307–323).
By the way, we will meet with the symmetric group S_3 again
when we return to take up the study of Peirce's early paper
"On a Class of Multiple Algebras" (CP 3.324-327), and also
his late unpublished work "The Simplest Mathematics" (1902)
(CP 4.227-323), with particular reference to the section
that treats of "Trichotomic Mathematics" (CP 4.307-323).
==Note 18==
==Note 18==