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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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, 17:10, 22 April 2012→6.6. Basic Notions of Group Theory: markup
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>
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>
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 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 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>.
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Table 32.1 Scheme of a Group Multiplication Table
Table 32.1 Scheme of a Group Multiplication Table
* x0 ... xj ...
* x0 ... xj ...