Changes

So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group.  This is a group of six elements, say, <math>G = \{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h}, \operatorname{i}, \operatorname{j} \},\!</math> with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, <math>X = \{ A, B, C \},\!</math> usually notated as <math>G = \operatorname{Sym}(X)</math> or more abstractly and briefly, as <math>\operatorname{Sym}(3)</math> or <math>S_3.\!</math>  The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in <math>\operatorname{Sym}(X).</math>

So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group.  This is a group of six elements, say, <math>G = \{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h}, \operatorname{i}, \operatorname{j} \},\!</math> with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, <math>X = \{ A, B, C \},\!</math> usually notated as <math>G = \operatorname{Sym}(X)</math> or more abstractly and briefly, as <math>\operatorname{Sym}(3)</math> or <math>S_3.\!</math>  The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in <math>\operatorname{Sym}(X).</math>

{| align="center" cellpadding="6" width="90%"

| align="center" |

 

<pre>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"

Table 2.  Permutations or Substitutions in Sym_{A, B, C}

|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>

o---------o---------o---------o---------o---------o---------o

|- style="background:#f0f0ff"

|         |         |         |         |         |        |

| width="16%" | <math>\operatorname{e}</math>

|   e    |   f    |   g    |   h    |   i    |   j   |

| width="16%" | <math>\operatorname{f}</math>

|         |        |        |        |        |        |

| width="16%" | <math>\operatorname{g}</math>

| width="16%" | <math>\operatorname{h}</math>

| width="16%" | <math>\operatorname{i}</math>

|  A B C | A B C |  A B C  | A B C |  A B C | A B C |

| width="16%" | <math>\operatorname{j}</math>

|-

|

|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |

|  A B C |  C A B |  B C A | A C B |  C B A |  B A C |

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\mathrm{A} & \mathrm{B} & \mathrm{C}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{C} & \mathrm{A} & \mathrm{B}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{B} & \mathrm{C} & \mathrm{A}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{C} & \mathrm{B} & \mathrm{A}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{B} & \mathrm{A} & \mathrm{C}

\end{matrix}</math>

|}

|}

Here is the operation table for <math>S_3,\!</math> given in abstract fashion:

Here is the operation table for <math>S_3,\!</math> given in abstract fashion:

| align="center" |

| align="center" |

<pre>

<pre>

Table 1.  Symmetric Group S_3

Symmetric Group S_3

o-------------------------------------------------o

o-------------------------------------------------o

|                                                |

|                                                |