Changes

===Note 16===

===Note 16===

'''Obstacle 2.''' Applying the pragmatic maxim, even with a moderate aim, can be hard. I think that my present example, deliberately impoverished as it is, affords us with an embarassing richness of evidence of just how complex the simple can be.

 

Obstacle 2.  Applying the pragmatic maxim, even with a moderate aim, can be hard.

I think that my present example, deliberately impoverished as it is, affords us

with an embarassing richness of evidence of just how complex the simple can be.

All the better reason for me to see if I can finish it up before moving on.

All the better reason for me to see if I can finish it up before moving on.

Expressed most simply, the idea is to replace the question of "what it is",

Expressed most simply, the idea is to replace the question of "what it is", which modest people know is far too difficult for them to answer right off, with the question of "what it does", which most of us know a modicum about.

which modest people know is far too difficult for them to answer right off,

with the question of "what it does", which most of us know a modicum about.

In the case of regular representations of groups we found

In the case of regular representations of groups we found a non-plussing surplus of answers to sort our way through. So let us track back one more time to see if we can learn any lessons that might carry over to more realistic cases.

a non-plussing surplus of answers to sort our way through.

So let us track back one more time to see if we can learn

any lessons that might carry over to more realistic cases.

Here is is the operation table of V_4 once again:

Here is is the operation table of V₄ once again:

Table 1.  Klein Four-Group V_4

Table 1.  Klein Four-Group V_4

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

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

|        %        |        |        |        |

|        %        |        |        |        |

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

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

A group operation table is really just a device for

A group operation table is really just a device for recording a certain 3-adic relation, to be specific, the set of triples of the form (''x'',&nbsp;''y'',&nbsp;''z'') satisfying the equation ''x''<math>\cdot</math>''y''&nbsp;=&nbsp;''z'', where "<math>\cdot</math>" signifies the group operation, usually omitted as understood in context.

recording a certain 3-adic relation, to be specific,

the set of triples of the form <x, y, z> satisfying

the equation x.y = z, where "." signifies the group

operation, usually omitted as understood in context.

In the case of V_4 = (G, .), where G is the "underlying set"

In the case of V₄ = (''G'',&nbsp;<math>\cdot</math>), where ''G'' is the "underlying set" {''e'',&nbsp;''f'',&nbsp;''g'',&nbsp;''h''}, we have the 3-adic relation ''L''(V₄)&nbsp;&sube;&nbsp;''G''&nbsp;&times;&nbsp;''G''&nbsp;&times;&nbsp;''G'' whose triples are listed below:

{e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G

whose triples are listed below:

   <e, e, e>

   <e, e, e>

   <e, f, f>

   <e, f, f>

   <h, g, f>

   <h, g, f>

   <h, h, e>

   <h, h, e>

It is part of the definition of a group that the 3-adic

It is part of the definition of a group that the 3-adic

relation L c G^3 is actually a function L : G x G -> G.

relation L c G^3 is actually a function L : G x G -> G.