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Tables 32.2 and 32.3 illustrate the two conceivable ways of forming a regular representation of a group <math>G.\!</math>
Tables 32.2 and 32.3 illustrate the two conceivable ways of forming a regular representation of a group <math>G.\!</math>
The '''ante-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 first operand of the group operation. Notating this function as <math>h_1(x_i) : G \to G,\!</math> the '''regular ante-representation''' of <math>G\!</math> is a map <math>h_1 : G \to (G \to G)\!</math> that is schematized in Table 32.2. Here, each of the functions <math>h_1(x_i) : G \to G\!</math> is represented as a set of ordered pairs of the form <math>(x_j ~,~ x_i * x_j).\!</math>
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The "post-representation" of xi in G is a function from G to G that is formed by considering the effects of xi on the elements of G when xi acts in the role of the second operand of the group operation. Notating this function as "h2(xi) : G >G", the "regular post representation" of G is a map h2 : G > (G >G) that is schematized in Table 32.3. Here, each of the functions h2(xi) : G >G is represented as a set of ordered pairs of the form <xj, xj*xi>.
The "post-representation" of xi in G is a function from G to G that is formed by considering the effects of xi on the elements of G when xi acts in the role of the second operand of the group operation. Notating this function as "h2(xi) : G >G", the "regular post representation" of G is a map h2 : G > (G >G) that is schematized in Table 32.3. Here, each of the functions h2(xi) : G >G is represented as a set of ordered pairs of the form <xj, xj*xi>.
