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, 19:56, 25 April 2012→6.6. Basic Notions of Group Theory: format table
This fact can be verified in several ways: (1) by checking that the map <math>h\!</math> is bijective and that <math>h(x \cdot y) = h(x) + h(y)\!</math> for every <math>x\!</math> and <math>y\!</math> in <math>Z_4(\cdot),\!</math> (2) by noting that <math>h\!</math> transforms the whole multiplication table for <math>Z_4(\cdot)\!</math> into the whole addition table for <math>Z_4(+)\!</math> in a one-to-one and onto fashion, or (3) by finding that both systems share some collection of properties that are definitive of the abstract group, for example, being cyclic of order <math>4.\!</math>
This fact can be verified in several ways: (1) by checking that the map <math>h\!</math> is bijective and that <math>h(x \cdot y) = h(x) + h(y)\!</math> for every <math>x\!</math> and <math>y\!</math> in <math>Z_4(\cdot),\!</math> (2) by noting that <math>h\!</math> transforms the whole multiplication table for <math>Z_4(\cdot)\!</math> into the whole addition table for <math>Z_4(+)\!</math> in a one-to-one and onto fashion, or (3) by finding that both systems share some collection of properties that are definitive of the abstract group, for example, being cyclic of order <math>4.\!</math>
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Table 34.1 Multiplicative Presentation of the Group Z4(.)
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|+ <math>\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)</math>
|- style="height:50px"
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
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| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{a}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{b}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{c}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
| <math>\operatorname{1}</math>
| <math>\operatorname{a}</math>
| <math>\operatorname{b}</math>
| <math>\operatorname{c}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{a}</math>
| <math>\operatorname{a}</math>
| <math>\operatorname{b}</math>
| <math>\operatorname{c}</math>
| <math>\operatorname{1}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{b}</math>
| <math>\operatorname{b}</math>
| <math>\operatorname{c}</math>
| <math>\operatorname{1}</math>
| <math>\operatorname{a}</math>
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{c}</math>
| <math>\operatorname{c}</math>
| <math>\operatorname{1}</math>
| <math>\operatorname{a}</math>
| <math>\operatorname{b}</math>
|}
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