Changes

==Note 23==

==Note 23==

| Bein' on the twenty-third of June,

|      As I sat weaving all at my loom,

| width="50%" |

| Bein' on the twenty-third of June,

Bein' on the twenty-third of June,<br>

|      As I sat weaving all at my loom,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;As I sat weaving all at my loom,<br>

| I heard a thrush, singing on yon bush,

Bein' on the twenty-third of June,<br>

|      And the song she sang was The Jug of Punch.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;As I sat weaving all at my loom,<br>

I heard a thrush, singing on yon bush,<br>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;And the song she sang was The Jug of Punch.<br>

We've seen a couple of groups, V_4 and S_3, represented in various ways, and

We've seen a couple of groups, <math>V_4\!</math> and <math>S_3,\!</math> represented in various ways, and we've seen their representations presented in a variety of different manners. Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called ''matrix representation'' of a group.

we've seen their representations presented in a variety of different manners.

Let us look at one other stylistic variant for presenting a representation

that is frequently seen, the so-called "matrix representation" of a group.

Recalling the manner of our acquaintance with the symmetric group S_3,

Recalling the manner of our acquaintance with the symmetric group <math>S_3,\!</math> we began with the ''bigraph'' (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set <math>X = \{ A, B, C \}.\!</math>

we began with the "bigraph" (bipartite graph) picture of its natural

representation as the set of all permutations or substitutions on

the set X = {A, B, C}.

Table 1.  Permutations or Substitutions in Sym {A, B, C}

Table 1.  Permutations or Substitutions in Sym {A, B, C}

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

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

|        |        |        |        |        |        |

|        |        |        |        |        |        |

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

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

Then we rewrote these permutations -- being functions f : X --> X

Then we rewrote these permutations -- being functions f : X --> X

they can also be recognized as being 2-adic relations f c X  x  X --

they can also be recognized as being 2-adic relations f c X  x  X --