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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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The next series of Tables presents the group operations and regular representations for the groups <math>V_4\!</math> and <math>Z_4.\!</math> If a group is abelian, as both of these groups are, then its <math>h_1\!</math> and <math>h_2\!</math> representations are indistinguishable, and a single form of regular representation <math>h : G \to (G \to G)\!</math> will do for both.
The next series of Tables presents the group operations and regular representations for the groups <math>V_4\!</math> and <math>Z_4.\!</math> If a group is abelian, as both of these groups are, then its <math>h_1\!</math> and <math>h_2\!</math> representations are indistinguishable, and a single form of regular representation <math>h : G \to (G \to G)\!</math> will do for both.
Table 33.1 shows the multiplication table of the group <math>V_4,\!</math> while Tables 33.2 and 33.3 present two versions of its regular representation. The first version, somewhat hastily, gives the functional representation of each group element as a set of ordered pairs of group elements. The second version, more circumspectly, gives the functional representative of each group element as a set of ordered pairs of element names, also referred to as ''objects'', ''points'', ''letters'', or ''symbols''.
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Table 33.1 Multiplication Operation of the Group V4
Table 33.1 Multiplication Operation of the Group V4
* 1 r s t
* 1 r s t
s s t 1 r
s s t 1 r
t t s r 1
t t s r 1
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Table 33.2 Regular Representation of the Group V4
Table 33.2 Regular Representation of the Group V4
Element Function as Set of Ordered Pairs of Elements
Element Function as Set of Ordered Pairs of Elements
s { <1, s>, <r, t>, <s, 1>, <t, r> }
s { <1, s>, <r, t>, <s, 1>, <t, r> }
t { <1, t>, <r, s>, <s, r>, <t, 1> }
t { <1, t>, <r, s>, <s, r>, <t, 1> }
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Table 33.3 Regular Representation of the Group V4
Table 33.3 Regular Representation of the Group V4
Element Function as Set of Ordered Pairs of Symbols
Element Function as Set of Ordered Pairs of Symbols
s { <"1", "s">, <"r", "t">, <"s", "1">, <"t", "r"> }
s { <"1", "s">, <"r", "t">, <"s", "1">, <"t", "r"> }
t { <"1", "t">, <"r", "s">, <"s", "r">, <"t", "1"> }
t { <"1", "t">, <"r", "s">, <"s", "r">, <"t", "1"> }
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Tables 34.1 and 35.1 show two forms of operation table for the group Z4, presenting the group, for the sake of contrast, in multiplicative and additive forms, respectively. Tables 34.2 and 35.2 give the corresponding forms of the regular representation.
Tables 34.1 and 35.1 show two forms of operation table for the group <math>Z_4,\!</math> presenting the group, for the sake of contrast, in multiplicative and additive forms, respectively. Tables 34.2 and 35.2 give the corresponding forms of the regular representation.
The multiplicative and additive versions of what is abstractly the same group, Z4, can be used to illustrate the concept of a group isomorphism.
The multiplicative and additive versions of what is abstractly the same group, <math>Z_4,\!</math> can be used to illustrate the concept of a group isomorphism.
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Let the multiplicative version of Z4 be formalized as:
Let the multiplicative version of Z4 be formalized as: