ONT Re: Differential Logic -- Series A
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note A17
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
So long as we're in the neighborhood, we might as well take in
some more of the sights, for instance, the smallest example of
a non-abelian (non-commutative) group. This is a group of six
elements, say, G = {e, f, g, h, i, j}, with no relation to any
other employment of these six symbols being implied, of course,
and it can be most easily represented as the permutation group
on a set of three letters, say, X = {A, B, C}, usually notated
as G = Sym(X) or more abstractly and briefly, as Sym(3) or S_3.
Here are the permutation (= substitution) operations in Sym(X):
Table 1. Permutations or Substitutions in Sym_{A, B, C}
o---------o---------o---------o---------o---------o---------o
| | | | | | |
| e | f | g | h | i | j |
| | | | | | |
o=========o=========o=========o=========o=========o=========o
| | | | | | |
| A B C | A B C | A B C | A B C | A B C | A B C |
| | | | | | |
| | | | | | | | | | | | | | | | | | | | | | | | |
| v v v | v v v | v v v | v v v | v v v | v v v |
| | | | | | |
| A B C | C A B | B C A | A C B | C B A | B A C |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
Here is the operation table for S_3, given in abstract fashion:
Table 2. Symmetric Group S_3
| ^
| e / \ e
| / \
| / e \
| f / \ / \ f
| / \ / \
| / f \ f \
| g / \ / \ / \ g
| / \ / \ / \
| / g \ g \ g \
| h / \ / \ / \ / \ h
| / \ / \ / \ / \
| / h \ e \ e \ h \
| i / \ / \ / \ / \ / \ i
| / \ / \ / \ / \ / \
| / i \ i \ f \ j \ i \
| j / \ / \ / \ / \ / \ / \ j
| / \ / \ / \ / \ / \ / \
| ( j \ j \ j \ i \ h \ j )
| \ / \ / \ / \ / \ / \ /
| \ / \ / \ / \ / \ / \ /
| \ h \ h \ e \ j \ i /
| \ / \ / \ / \ / \ /
| \ / \ / \ / \ / \ /
| \ i \ g \ f \ h /
| \ / \ / \ / \ /
| \ / \ / \ / \ /
| \ f \ e \ g /
| \ / \ / \ /
| \ / \ / \ /
| \ g \ f /
| \ / \ /
| \ / \ /
| \ e /
| \ /
| \ /
| v
By the way, we will meet with the symmetric group S_3 again
when we return to take up the study of Peirce's early paper
"On a Class of Multiple Algebras" (CP 3.324-327), and also
his late unpublished work "The Simplest Mathematics" (1902)
(CP 4.227-323), with particular reference to the section
that treats of "Trichotomic Mathematics" (CP 4.307-323).
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
http://www.cs.bsu.edu/homepages/mighty/history.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o