Changes

<br>

<br>

<pre>

Table 33.3 Regular Representation of the Group V4

|+ <math>\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4</math>

Element Function as Set of Ordered Pairs of Symbols

1   { <"1", "1">, <"r", "r">, <"s", "s">, <"t", "t"> }

| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>

r   { <"1", "r">, <"r", "1">, <"s", "t">, <"t", "s"> }

| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math>

s   { <"1", "s">, <"r", "t">, <"s", "1">, <"t", "r"> }

t   { <"1", "t">, <"r", "s">, <"s", "r">, <"t", "1"> }

| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>

</pre>

| 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>

Tables&nbsp;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&nbsp;34.2 and 35.2 give the corresponding forms of the regular representation.

Tables&nbsp;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&nbsp;34.2 and 35.2 give the corresponding forms of the regular representation.