Changes

By way of collecting a short-term pay-off for all the work &mdash; not to mention all the peirce-spiration &mdash; 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 &mdash; not to mention all the peirce-spiration &mdash; 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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<pre>

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Table 1.  Permutations or Substitutions in Sym {A, B, C}

|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>

o---------o---------o---------o---------o---------o---------o

|- style="background:#f0f0ff"

|         |         |         |         |         |        |

| width="16%" | <math>\operatorname{e}</math>

|   e    |   f    |   g    |   h    |   i    |   j   |

| width="16%" | <math>\operatorname{f}</math>

|         |        |        |        |        |        |

| width="16%" | <math>\operatorname{g}</math>

| width="16%" | <math>\operatorname{h}</math>

| width="16%" | <math>\operatorname{i}</math>

|  A B C | A B C |  A B C  | A B C |  A B C | A B C |

| width="16%" | <math>\operatorname{j}</math>

|-

|

|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |

|  A B C |  C A B |  B C A | A C B |  C B A |  B A C |

</pre>

\mathrm{A} & \mathrm{B} & \mathrm{C}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{C} & \mathrm{A} & \mathrm{B}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{B} & \mathrm{C} & \mathrm{A}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{C} & \mathrm{B} & \mathrm{A}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{B} & \mathrm{A} & \mathrm{C}

\end{matrix}</math>

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|}

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>