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User:Jon Awbrey/TABLE (view source)

Revision as of 03:22, 26 April 2012

10,834 bytes added ,  03:22, 26 April 2012
→‎Group Operation Tables
Line 6,457: Line 6,457:  
|+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>
 
|+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>
 
|- style="height:50px"
 
|- style="height:50px"
−
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math>
+
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
−
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
+
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math>
−
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{r}</math>
+
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math>
−
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{s}</math>
+
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math>
−
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{t}</math>
+
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math>
−
|- style="height:50px"
  −
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
  −
| <math>\operatorname{1}</math>
  −
| <math>\operatorname{r}</math>
  −
| <math>\operatorname{s}</math>
  −
| <math>\operatorname{t}</math>
  −
|- style="height:50px"
  −
| style="border-right:1px solid black" | <math>\operatorname{r}</math>
  −
| <math>\operatorname{r}</math>
  −
| <math>\operatorname{1}</math>
  −
| <math>\operatorname{t}</math>
  −
| <math>\operatorname{s}</math>
  −
|- style="height:50px"
  −
| style="border-right:1px solid black" | <math>\operatorname{s}</math>
  −
| <math>\operatorname{s}</math>
  −
| <math>\operatorname{t}</math>
  −
| <math>\operatorname{1}</math>
  −
| <math>\operatorname{r}</math>
  −
|- style="height:50px"
  −
| style="border-right:1px solid black" | <math>\operatorname{t}</math>
  −
| <math>\operatorname{t}</math>
  −
| <math>\operatorname{s}</math>
  −
| <math>\operatorname{r}</math>
  −
| <math>\operatorname{1}</math>
  −
|}
  −
 
  −
<br>
  −
 
  −
{| 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 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>
  −
|- style="height:50px"
  −
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
  −
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math>
  −
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math>
  −
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math>
  −
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math>
   
|- style="height:50px"
 
|- style="height:50px"
 
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
 
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
Line 6,522: Line 6,486:  
| <math>\operatorname{f}</math>
 
| <math>\operatorname{f}</math>
 
| <math>\operatorname{e}</math>
 
| <math>\operatorname{e}</math>
 +
|}
 +
 +
<br>
 +
 +
{| 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 33.2}~~\text{Regular Representation of the Group}~V_4</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="16%" | <math>(\operatorname{e}, \operatorname{e}),</math>
 +
| width="20%" | <math>(\operatorname{f}, \operatorname{f}),</math>
 +
| width="20%" | <math>(\operatorname{g}, \operatorname{g}),</math>
 +
| width="16%" | <math>(\operatorname{h}, \operatorname{h})</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{e}, \operatorname{f}),</math>
 +
| <math>(\operatorname{f}, \operatorname{e}),</math>
 +
| <math>(\operatorname{g}, \operatorname{h}),</math>
 +
| <math>(\operatorname{h}, \operatorname{g})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{e}, \operatorname{g}),</math>
 +
| <math>(\operatorname{f}, \operatorname{h}),</math>
 +
| <math>(\operatorname{g}, \operatorname{e}),</math>
 +
| <math>(\operatorname{h}, \operatorname{f})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{e}, \operatorname{h}),</math>
 +
| <math>(\operatorname{f}, \operatorname{g}),</math>
 +
| <math>(\operatorname{g}, \operatorname{f}),</math>
 +
| <math>(\operatorname{h}, \operatorname{e})</math>
 +
| <math>\}\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| 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 33.3}~~\text{Regular Representation of the Group}~V_4</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
 +
| width="20%" | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
 +
| width="20%" | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
 +
| width="16%" | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\{\!</math>
 +
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\{\!</math>
 +
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\{\!</math>
 +
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})</math>
 +
| <math>\}\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| 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>
 +
| 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>
 +
|}
 +
 +
<br>
 +
 +
{| 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 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-right:1px solid black" | <math>\operatorname{1}</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="16%" | <math>(\operatorname{1}, \operatorname{1}),</math>
 +
| width="20%" | <math>(\operatorname{a}, \operatorname{a}),</math>
 +
| width="20%" | <math>(\operatorname{b}, \operatorname{b}),</math>
 +
| width="16%" | <math>(\operatorname{c}, \operatorname{c})</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{a}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{1}, \operatorname{a}),</math>
 +
| <math>(\operatorname{a}, \operatorname{b}),</math>
 +
| <math>(\operatorname{b}, \operatorname{c}),</math>
 +
| <math>(\operatorname{c}, \operatorname{1})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{b}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{1}, \operatorname{b}),</math>
 +
| <math>(\operatorname{a}, \operatorname{c}),</math>
 +
| <math>(\operatorname{b}, \operatorname{1}),</math>
 +
| <math>(\operatorname{c}, \operatorname{a})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{c}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{1}, \operatorname{c}),</math>
 +
| <math>(\operatorname{a}, \operatorname{1}),</math>
 +
| <math>(\operatorname{b}, \operatorname{a}),</math>
 +
| <math>(\operatorname{c}, \operatorname{b})</math>
 +
| <math>\}\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| 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 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{0}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{0}</math>
 +
| <math>\operatorname{0}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{2}</math>
 +
| <math>\operatorname{3}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{2}</math>
 +
| <math>\operatorname{3}</math>
 +
| <math>\operatorname{0}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{2}</math>
 +
| <math>\operatorname{2}</math>
 +
| <math>\operatorname{3}</math>
 +
| <math>\operatorname{0}</math>
 +
| <math>\operatorname{1}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{3}</math>
 +
| <math>\operatorname{3}</math>
 +
| <math>\operatorname{0}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{2}</math>
 +
|}
 +
 +
<br>
 +
 +
{| 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 35.2}~~\text{Regular Representation of the Group}~Z_4(+)</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-right:1px solid black" | <math>\operatorname{0}</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="16%" | <math>(\operatorname{0}, \operatorname{0}),</math>
 +
| width="20%" | <math>(\operatorname{1}, \operatorname{1}),</math>
 +
| width="20%" | <math>(\operatorname{2}, \operatorname{2}),</math>
 +
| width="16%" | <math>(\operatorname{3}, \operatorname{3})</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{0}, \operatorname{1}),</math>
 +
| <math>(\operatorname{1}, \operatorname{2}),</math>
 +
| <math>(\operatorname{2}, \operatorname{3}),</math>
 +
| <math>(\operatorname{3}, \operatorname{0})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{2}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{0}, \operatorname{2}),</math>
 +
| <math>(\operatorname{1}, \operatorname{3}),</math>
 +
| <math>(\operatorname{2}, \operatorname{0}),</math>
 +
| <math>(\operatorname{3}, \operatorname{1})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{3}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{0}, \operatorname{3}),</math>
 +
| <math>(\operatorname{1}, \operatorname{0}),</math>
 +
| <math>(\operatorname{2}, \operatorname{1}),</math>
 +
| <math>(\operatorname{3}, \operatorname{2})</math>
 +
| <math>\}\!</math>
 
|}
 
|}
  
Jon Awbrey
12,080

edits

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