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==Note 16==
 
==Note 16==
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<pre>
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We've been exploring the applications of a certain technique for clarifying abstruse concepts, a rough-cut version of the pragmatic maxim that I've been accustomed to refer to as the ''operationalization'' of ideas.  The basic idea is to replace the question of ''What it is'', which modest people comprehend is far beyond their powers to answer definitively any time soon, with the question of ''What it does'', which most people know at least a modicum about.
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We've been exploring the applications of a certain technique
  −
for clarifying abstruse concepts, a rough-cut version of the
  −
pragmatic maxim that I've been accustomed to refer to as the
  −
"operationalization" of ideas.  The basic idea is to replace
  −
the question of "What it is", which modest people comprehend
  −
is far beyond their powers to answer any time soon, with the
  −
question of "What it does", which most people know at least
  −
a modicum about.
     −
In the case of regular representations of groups we found
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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.
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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
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any lessons that might carry over to more realistic cases.
     −
Here is is the operation table of V_4 once again:
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Here is is the operation table of <math>V_4\!</math> once again:
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o-------o-------o-------o-------o-------o
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<br>
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|      %      |      |      |      |
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|  *  %  e  |  f  |  g  |  h  |
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|      %      |      |      |      |
  −
o=======o=======o=======o=======o=======o
  −
|      %      |      |      |      |
  −
|  e  %  e  |  f  |  g  |  h  |
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|      %      |      |      |      |
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o-------o-------o-------o-------o-------o
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|      %      |      |      |      |
  −
|  f  %  f  |  e  |  h  |  g  |
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|      %      |      |      |      |
  −
o-------o-------o-------o-------o-------o
  −
|      %      |      |      |      |
  −
|  g  %  g  |  h  |  e  |  f  |
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|      %      |      |      |      |
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o-------o-------o-------o-------o-------o
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|      %      |      |      |      |
  −
|  h  %  h  |  g  |  f  |  e  |
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|      %      |      |      |      |
  −
o-------o-------o-------o-------o-------o
     −
A group operation table is really just a device for recording
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{| 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%"
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a certain 3-adic relation, specifically, the set of 3-tuples
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|+ <math>\text{Klein Four-Group}~ V_4</math>
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of the form <x, y, z> that satisfy the equation x * y = z,
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|- style="height:50px"
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where the sign '*' that indicates the group operation is
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| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
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frequently omitted in contexts where it is understood.
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| width="22%" style="border-bottom:1px solid black" |
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<math>\operatorname{e}</math>
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| width="22%" style="border-bottom:1px solid black" |
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<math>\operatorname{f}</math>
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| width="22%" style="border-bottom:1px solid black" |
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<math>\operatorname{g}</math>
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| width="22%" style="border-bottom:1px solid black" |
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<math>\operatorname{h}</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\operatorname{e}</math>
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| <math>\operatorname{e}</math>
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| <math>\operatorname{f}</math>
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| <math>\operatorname{g}</math>
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| <math>\operatorname{h}</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\operatorname{f}</math>
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| <math>\operatorname{f}</math>
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| <math>\operatorname{e}</math>
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| <math>\operatorname{h}</math>
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| <math>\operatorname{g}</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\operatorname{g}</math>
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| <math>\operatorname{g}</math>
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| <math>\operatorname{h}</math>
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| <math>\operatorname{e}</math>
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| <math>\operatorname{f}</math>
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|- style="height:50px"
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| style="border-right:1px solid black" | <math>\operatorname{h}</math>
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| <math>\operatorname{h}</math>
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| <math>\operatorname{g}</math>
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| <math>\operatorname{f}</math>
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| <math>\operatorname{e}</math>
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|}
   −
In the case of V_4 = (G, *), where G is the "underlying set"
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<br>
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{e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G
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whose triples are listed below:
     −
  e:e:e
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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 <math>(x, y, z)\!</math> satisfying the equation <math>x \cdot y = z.</math>
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  e:f:f
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  e:g:g
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  e:h:h
     −
  f:e:f
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In the case of <math>V_4 = (G, \cdot),</math> where <math>G\!</math> is the ''underlying set'' <math>\{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h} \},</math> we have the 3-adic relation <math>L(V_4) \subseteq G \times G \times G</math> whose triples are listed below:
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  f:f:e
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  f:g:h
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  f:h:g
     −
  g:e:g
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{| align="center" cellpadding="6" width="90%"
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  g:f:h
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| align="center" |
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  g:g:e
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<math>\begin{matrix}
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  g:h:f
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(\operatorname{e}, \operatorname{e}, \operatorname{e}) &
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(\operatorname{e}, \operatorname{f}, \operatorname{f}) &
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  h:e:h
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(\operatorname{e}, \operatorname{g}, \operatorname{g}) &
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  h:f:g
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(\operatorname{e}, \operatorname{h}, \operatorname{h})
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  h:g:f
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\\[6pt]
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  h:h:e
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(\operatorname{f}, \operatorname{e}, \operatorname{f}) &
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(\operatorname{f}, \operatorname{f}, \operatorname{e}) &
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(\operatorname{f}, \operatorname{g}, \operatorname{h}) &
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(\operatorname{f}, \operatorname{h}, \operatorname{g})
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\\[6pt]
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(\operatorname{g}, \operatorname{e}, \operatorname{g}) &
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(\operatorname{g}, \operatorname{f}, \operatorname{h}) &
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(\operatorname{g}, \operatorname{g}, \operatorname{e}) &
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(\operatorname{g}, \operatorname{h}, \operatorname{f})
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\\[6pt]
 +
(\operatorname{h}, \operatorname{e}, \operatorname{h}) &
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(\operatorname{h}, \operatorname{f}, \operatorname{g}) &
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(\operatorname{h}, \operatorname{g}, \operatorname{f}) &
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(\operatorname{h}, \operatorname{h}, \operatorname{e})
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\end{matrix}</math>
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|}
    +
<pre>
 
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.
Jon Awbrey
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