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The '''post-representation''' of <math>x_i\!</math> in <math>G\!</math> is a function from <math>G\!</math> to <math>G\!</math> that is formed by considering the effects of <math>x_i\!</math> on the elements of <math>G\!</math> when <math>x_i\!</math> acts in the role of the second operand of the group operation.  Notating this function as <math>h_2(x_i) : G \to G,\!</math> the '''regular post-representation''' of <math>G\!</math> is a map <math>h_2 : G \to (G \to G)\!</math> that is schematized in Table&nbsp;32.3.  Here, each of the functions <math>h_2(x_i) : G \to G\!</math> is represented as a set of ordered pairs of the form <math>(x_j ~,~ x_j * x_i).\!</math>

The '''post-representation''' of <math>x_i\!</math> in <math>G\!</math> is a function from <math>G\!</math> to <math>G\!</math> that is formed by considering the effects of <math>x_i\!</math> on the elements of <math>G\!</math> when <math>x_i\!</math> acts in the role of the second operand of the group operation.  Notating this function as <math>h_2(x_i) : G \to G,\!</math> the '''regular post-representation''' of <math>G\!</math> is a map <math>h_2 : G \to (G \to G)\!</math> that is schematized in Table&nbsp;32.3.  Here, each of the functions <math>h_2(x_i) : G \to G\!</math> is represented as a set of ordered pairs of the form <math>(x_j ~,~ x_j * x_i).\!</math>

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Table 32.1 Scheme of a Group Multiplication Table

 

* x0 ... xj ...

x0 x0*x0 ... x0*xj ...

|+ <math>\text{Table 32.1}~~\text{Scheme of a Group Operation Table}</math>

xi xi*x0 ... xi*xj ...

| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math>

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| <math>x_0 * x_0\!</math>

| <math>x_0 * x_j\!</math>

| <math>x_i * x_0\!</math>

| <math>x_i * x_j\!</math>

| <math>\ldots\!</math>

 

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In following these maps, notice how closely one is treading in these representations to defining each element in terms of itself, but without quite going that far.  There are a couple of catches that save this form of representation from falling into a &ldquo;vicious circle&rdquo;, that is, into a pattern of self-reference that would beg the question of a definition and vitiate its usefulness as an explanation of each group element's action.  First, the regular representations do not represent that a group element is literally ''equal to'' a set of ordered pairs involving that very same group element, but only that it is ''mapped to'' something like this set.  Second, careful usage would dictate that the ''something like'' that one finds in the image of a representation, being something that is specified only up to its isomorphism class, is a transformation that really acts, not on the group elements <math>x_j\!</math> themselves, but only on their inert tokens, inactive images, partial symbols, passing names, or transitory signs of the form <math>{}^{\backprime\backprime} x_j {}^{\prime\prime}.\!</math>

In following these maps, notice how closely one is treading in these representations to defining each element in terms of itself, but without quite going that far.  There are a couple of catches that save this form of representation from falling into a &ldquo;vicious circle&rdquo;, that is, into a pattern of self-reference that would beg the question of a definition and vitiate its usefulness as an explanation of each group element's action.  First, the regular representations do not represent that a group element is literally ''equal to'' a set of ordered pairs involving that very same group element, but only that it is ''mapped to'' something like this set.  Second, careful usage would dictate that the ''something like'' that one finds in the image of a representation, being something that is specified only up to its isomorphism class, is a transformation that really acts, not on the group elements <math>x_j\!</math> themselves, but only on their inert tokens, inactive images, partial symbols, passing names, or transitory signs of the form <math>{}^{\backprime\backprime} x_j {}^{\prime\prime}.\!</math>