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By way of collecting a short-term pay-off for all the work — not to mention all the peirce-spiration — that we sweated out over the regular representations of the Klein 4-group <math>V_4,\!</math> let us write out as quickly as possible in ''relative form'' a minimal budget of representations of the symmetric group on three letters, <math>S_3 = \operatorname{Sym}(3).</math> After doing the usual bit of compare and contrast among these divers representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscur'd details of Peirce's early "Algebra + Logic" papers.
By way of collecting a short-term pay-off for all the work — not to mention all the peirce-spiration — that we sweated out over the regular representations of the Klein 4-group <math>V_4,\!</math> let us write out as quickly as possible in ''relative form'' a minimal budget of representations of the symmetric group on three letters, <math>S_3 = \operatorname{Sym}(3).</math> After doing the usual bit of compare and contrast among these divers representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscur'd details of Peirce's early "Algebra + Logic" papers.
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{| align="center" cellpadding="10" width="90%"
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<pre>
<pre>
Writing this table in relative form generates the following natural representation of <math>S_3.\!</math>
Writing this table in relative form generates the following natural representation of <math>S_3.\!</math>
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="10" width="90%"
|
| align="center" |
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{e}
\operatorname{e}
& = & \operatorname{A}:\operatorname{A}
& = & \operatorname{A}\!:\!\operatorname{A}
& + & \operatorname{B}:\operatorname{B}
& + & \operatorname{B}\!:\!\operatorname{B}
& + & \operatorname{C}:\operatorname{C}
& + & \operatorname{C}\!:\!\operatorname{C}
\\[4pt]
\\[4pt]
\operatorname{f}
\operatorname{f}
& = & \operatorname{A}:\operatorname{C}
& = & \operatorname{A}\!:\!\operatorname{C}
& + & \operatorname{B}:\operatorname{A}
& + & \operatorname{B}\!:\!\operatorname{A}
& + & \operatorname{C}:\operatorname{B}
& + & \operatorname{C}\!:\!\operatorname{B}
\\[4pt]
\\[4pt]
\operatorname{g}
\operatorname{g}
& = & \operatorname{A}:\operatorname{B}
& = & \operatorname{A}\!:\!\operatorname{B}
& + & \operatorname{B}:\operatorname{C}
& + & \operatorname{B}\!:\!\operatorname{C}
& + & \operatorname{C}:\operatorname{A}
& + & \operatorname{C}\!:\!\operatorname{A}
\\[4pt]
\\[4pt]
\operatorname{h}
\operatorname{h}
& = & \operatorname{A}:\operatorname{A}
& = & \operatorname{A}\!:\!\operatorname{A}
& + & \operatorname{B}:\operatorname{C}
& + & \operatorname{B}\!:\!\operatorname{C}
& + & \operatorname{C}:\operatorname{B}
& + & \operatorname{C}\!:\!\operatorname{B}
\\[4pt]
\\[4pt]
\operatorname{i}
\operatorname{i}
& = & \operatorname{A}:\operatorname{C}
& = & \operatorname{A}\!:\!\operatorname{C}
& + & \operatorname{B}:\operatorname{B}
& + & \operatorname{B}\!:\!\operatorname{B}
& + & \operatorname{C}:\operatorname{A}
& + & \operatorname{C}\!:\!\operatorname{A}
\\[4pt]
\\[4pt]
\operatorname{j}
\operatorname{j}
& = & \operatorname{A}:\operatorname{B}
& = & \operatorname{A}\!:\!\operatorname{B}
& + & \operatorname{B}:\operatorname{A}
& + & \operatorname{B}\!:\!\operatorname{A}
& + & \operatorname{C}:\operatorname{C}
& + & \operatorname{C}\!:\!\operatorname{C}
\end{matrix}</math>
\end{matrix}</math>
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