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MyWikiBiz, Author Your Legacy — Monday December 23, 2024
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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 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>
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<pre>
 
<pre>
 
Table 2.  Permutations or Substitutions in Sym_{A, B, C}
 
Table 2.  Permutations or Substitutions in Sym_{A, B, C}
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o---------o---------o---------o---------o---------o---------o
 
o---------o---------o---------o---------o---------o---------o
 
</pre>
 
</pre>
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Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
 
Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
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<pre>
 
<pre>
Table 3.  Symmetric Group S_3
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Table 1.  Symmetric Group S_3
 
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o-------------------------------------------------o
|                        _
+
|                                                 |
|                    e / \ e
+
|                        ^                       |
|                      /  \
+
|                    e / \ e                     |
|                    /  e  \
+
|                      /  \                     |
|                  f / \  / \ f
+
|                    /  e  \                     |
|                  /  \ /  \
+
|                  f / \  / \ f                 |
|                  /  f  \  f  \
+
|                  /  \ /  \                   |
|              g / \  / \  / \ g
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|                  /  f  \  f  \                 |
|                /  \ /  \ /  \
+
|              g / \  / \  / \ g               |
|              /  g  \  g  \  g  \
+
|                /  \ /  \ /  \               |
|            h / \  / \  / \  / \ h
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|              /  g  \  g  \  g  \               |
|            /  \ /  \ /  \ /  \
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|            h / \  / \  / \  / \ h           |
|            /  h  \  e  \  e  \  h  \
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|            /  \ /  \ /  \ /  \             |
|        i / \  / \  / \  / \  / \ i
+
|            /  h  \  e  \  e  \  h  \           |
|          /  \ /  \ /  \ /  \ /  \
+
|        i / \  / \  / \  / \  / \ i         |
|        /  i  \  i  \  f  \  j  \  i  \
+
|          /  \ /  \ /  \ /  \ /  \         |
|      j / \  / \  / \  / \  / \  / \ j
+
|        /  i  \  i  \  f  \  j  \  i  \         |
|      /  \ /  \ /  \ /  \ /  \ /  \
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|      j / \  / \  / \  / \  / \  / \ j     |
|      (  j  \  j  \  j  \  i  \  h  \  j  )
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|      /  \ /  \ /  \ /  \ /  \ /  \       |
|      \  / \  / \  / \  / \  / \  /
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|      (  j  \  j  \  j  \  i  \  h  \  j  )     |
|        \ /  \ /  \ /  \ /  \ /  \ /
+
|      \  / \  / \  / \  / \  / \  /       |
|        \  h  \  h  \  e  \  j  \  i  /
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|        \ /  \ /  \ /  \ /  \ /  \ /       |
|          \  / \  / \  / \  / \  /
+
|        \  h  \  h  \  e  \  j  \  i  /         |
|          \ /  \ /  \ /  \ /  \ /
+
|          \  / \  / \  / \  / \  /         |
|            \  i  \  g  \  f  \  h  /
+
|          \ /  \ /  \ /  \ /  \ /           |
|            \  / \  / \  / \  /
+
|            \  i  \  g  \  f  \  h  /           |
|              \ /  \ /  \ /  \ /
+
|            \  / \  / \  / \  /             |
|              \  f  \  e  \  g  /
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|              \ /  \ /  \ /  \ /             |
|                \  / \  / \  /
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|              \  f  \  e  \  g  /               |
|                \ /  \ /  \ /
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|                \  / \  / \  /               |
|                  \  g  \  f  /
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|                \ /  \ /  \ /                 |
|                  \  / \  /
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|                  \  g  \  f  /                 |
|                    \ /  \ /
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|                  \  / \  /                   |
|                    \  e  /
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|                    \ /  \ /                   |
|                      \  /
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|                    \  e  /                     |
|                      \ /
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|                      \  /                     |
|                        ¯
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|                      \ /                       |
 +
|                        v                        |
 +
|                                                |
 +
o-------------------------------------------------o
 
</pre>
 
</pre>
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|}
    
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&ndash;327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227&ndash;323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307&ndash;323).
 
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&ndash;327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227&ndash;323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307&ndash;323).
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