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Symmetric Groups
----------------
S_{4} = permutations of {1,2,3,4}
|S_{n}| = n!
∴ |S_{4}| = 24 elements.
These are 1234, 1342, 4321 etc. etc.
Permutations can be regarded as the map,
f: {1,2,3,4} -> {1,2,3,4}
Where f is a function.
Treating the permutations as functions allows us
to define multiplicatio.
Consider f: 1234 -> 1342 and g: 1234 -> 4321
The corresponding functions are:
f(1) = 1, f(2) = 3, f(3) = 4, f(4) = 2
g(1) = 4, g(2) = 3, g(3) = 2, g(4) = 1
The compositions are:
f o g(1) = 2,
f o g(2) = 4
f o g(3) = 3
f o g(4) = 1
This is sometimes written in shorthand as:
- - - - - -
| 1 2 3 4 | . | 1 2 3 4 | = | 1 2 3 4 |
| 1 3 4 2 | | 4 3 2 1 | | 2 4 3 1 |
- - - - - -
The identity permutation is:
- -
| 1 2 3 4 |
| 1 2 3 4 |
- -
The inverses can be obtained by interchanging
the 2 rows (and sorting the order). For example,
- - - - - -
| 1 2 3 4 |^{-1} = | 1 3 4 2 | = | 1 2 3 4 |
| 1 3 4 2 |^{ } | 1 2 3 4 | | 1 4 2 3 |
- - - - - -
Therefore,
- - - - - -
| 1 2 3 4 | . | 1 2 3 4 | = | 1 2 3 4 |
| 1 3 4 2 | | 1 4 2 3 | | 1 2 3 4 |
- - - - - -
Multiplication enables us to construct a Cayley
table as follows:
x | 1234 | 4321 | ....
-----+------+------+------
1234 | 1234 | 4321 | ....
-----+------+------+------
1342 | 1342 | 2431 | ....
-----+------+------+------
.... | .... | .... | ....
Completing the entire table shows that the closure
property is satisfied.
Cycle Notation
--------------
Another notation used is cycle notation.
Consider:
- -
| 1 2 3 4 5 6 7 8 |
| 5 4 7 6 2 8 3 1 |
- -
(1 -> 5 -> 2 -> 4 -> 6 -> 8 -> 1 and 3-> 7
(1 5 2 4 6 8)(3 7)
Cayley's Theorem
----------------
Every finite group is isomorphic to a subgroup
of the symmetric group.
Cayley's theorem can be understood as a group
action on itself whereby multiplying an element
on the left by another element produces another
element. We can think of the ith row of the
Cayley table as defining a function, f_{i} which
takes j to i.j. This function can be seen as
a permutation of the elements of the group
and as a element of the symmetric group.
Example:
Consider the symmetry group of the square, D_{4}:
D_{4} = {R_{0},R_{90},R_{180},R_{270},m_{1},m_{2},δ_{1},δ_{2}}
R_{0} = 1234
R_{90} = 2341
R_{180} = 3412
R_{270} = 4123
m_{1} = 2143
m_{2} = 4321
δ_{1} = 3214
δ_{2} = 1432
The corresponding Cayley table (partial) is:
x | 1234 | 2341 | 2143 ...
-----+------+------+------
1234 | 1234 | 2341 | 2143 ...
-----+------+------+------
2341 | 2341 | 3412 | 3214 ...
-----+------+------+------
2143 | 2143 | 1432 | 1234 ...
. . . .
. . . .
. . . .
If the entire table is completed it is easy to
see that each row is a realization of D_{4} as a
subgroup of S_{4}.