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Revision as of 19:45, 23 April 2008
, 19:45, 23 April 2008→Note 19: markup
===Note 19===
===Note 19===
To construct the regular representations of ''S''<sub>3</sub>, we pick up from the data of its operation table:
<pre>
<pre>
Table 1. Symmetric Group S_3
Table 1. Symmetric Group S_3
| \ /
| \ /
| v
| v
</pre>
Just by way of staying clear about what we are doing,
Just by way of staying clear about what we are doing, let's return to the recipe that we worked out before:
let's return to the recipe that we worked out before:
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'' ⊆ ''G''<sup>3</sup> is actually a function ''L'' : ''G'' × ''G'' → ''G''. It is from this functional perspective that we can see an easy way to derive the two regular representations.
relation L c G^3 is actually a function L : G x G -> G.
It is from this functional perspective that we can see
an easy way to derive the two regular representations.
<pre>
Since we have a function of the type L : G x G -> G,
Since we have a function of the type L : G x G -> G,
we can define a couple of substitution operators:
we can define a couple of substitution operators:
