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Revision as of 19:45, 23 April 2008
, 19:45, 23 April 2008→Note 19: markup
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To construct the regular representations of S_3,
−we pick up from the data of its operation table:
−
===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
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| \ /
| \ /
| 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:
