Automorphism
Terminology
- Aut(G) : automorphism : G -> G ; isomorphism
- Int(G) : inner automorphism : G -> G where G -> Int(G) ; homomorphism
- Out(G) : outer automorphism := Aut(G) \ Int(G)
# $Aut(A_4)$ $\sim$ $S_4$
imagine the rotations of a tetrahedron to itself. Each rotation moves a point to the position which another point was. pick up any 2 faces and name a, b as 120˚ anti-clockwise rotations of each. (so, the above picture is clockwise i.e. it is mirrored and do not mind that.)
$ A _ 4 $ is described only with a, b and their conpositions as id = aaa = bbb.
$ A _ 4 $ = {
~~~
id,
aab, aba, baa,
a, aa,
b, bb,
abab, ab,
baba, ba,
~~~
}
if we define `c := abab`, and `d := baba`, then;
$ A _4 $ = {
~~~
id,
aab, aba, baa, <= 180˚ rotations
a, aa,
b, bb,
c, cc,
d, dd
~~~
}
where
~~~
a, b, c, d is 120˚ anti-clockwise
aa, bb, cc, dd is 240˚anti-clockwise or 120˚ clockwise
~~~
A member of Aut( $ A _ 4 $ ) should be isomorphism i.e. homomorphic, so it maps `a` to the 120˚ rotations. There is ;
~~~
a |-> a,b,c,d,aa,bb,cc,dd (8 patterns)
~~~
if we choose `bb` as the target,
~~~
a |-> bb
~~~
then because of homomorphism, `aa` is determined like,
~~~
aa |-> bbbb == b
~~~
Next, determine the target of `b`
Because this function `a` $ \mapsto $ `bb` is `anti-clockwise` $ \mapsto $ `clockwise`
`b` satisfies;
~~~
b |-> clockwise
~~~
so,
~~~
b |-> aa,cc,dd (3 patterns)
~~~
so Aut( $ A _ 4 $ ) has 8 x 3 = 24 elements.
but why is this isomorphic to $ S _ 4 $?
# Outer Automorphisms
$ S _ 4 $ is said to be $ G( T _ 6 ) $, as $ A _ 4 $ to $ G( T _ 4) $.
This means we can regard $ S _ 4 $ as the rotaions of a cube to its self.
A cube has at least one tetrahedron inside of itself as the above picture.
It means it exactly includes $ A _ 4 $ visibly.
Then go back to Aut( $ A _ 4 $ ) .
Aut ( $ A _ 4 $ ) include $ A _ 4 $ itself as Int( $ A _ 4 $ ).
Aut ( $ A _ 4 $ ) consists of 24 elements, and $ A _ 4 $ has 12.
So there is the other 12 elements, which we call Out( $ A _ 4 $ ).
Out( $ A _ 4 $ ) is consists of members which is not $ A _ 4 $, e.g. a mirror projection can map $ A _ 4 $ to itself.
In the above picture `z` $ \in $ $ S _ 4 $ and we can produce a mirror projection from `z`;
` ι(z)(-) = z * - * z` $ {} ^ {-1} $
z is 180 rotation and ι(z) is a mirror projection about xy-plain!
(Note the above picture is quite misleading. ι(z) is not xz-plain reflection. )
If you rotate 180 degree with z axix fixed, then some operation and again rotate 180 degree with z fixed. It generates exactly its mirrored operation!
As the same way, this `ι(-)` maps other elements of $ S _ 4 \setminus A _ 4 $ to Out( $ A _ 4 $ ). And this function is an isomorphism between $ S _ 4 $ and Aut( $ A _ 4 $ ).
# Dihedral groups $D_n$ are also interesting.
There is interesting facts;
~~~
Int(D3) == Aut(D3)
Int(D5) != Aut(D5)
~~~
assume 'a' is a (360˚/n) clockwise rotation and 'b' is a 180˚flipping rotation
then consider Automorphism in the same way;
~~~
b |-> b, ba, ... , ba^{n-1} (n patterns)
~~~
then
~~~
a |-> ?
~~~
since every member of Int(D5) is described as a D5 member,
the codomain is clockwise or anti-clockwise (not a 180˚flipping rotation because of homomorphism).
~~~
Int(D5)
72˚ -> 72˚ (a |-> a) (anti-clockwise -> anti-clockwise),
72˚ -> -72˚ (a |-> aaaa) (anti-clockwise -> clockwise)
Aut(D5)
72˚ -> 72˚,
72˚ -> 144˚, (a |-> aa) (anti-clockwise -> butterfly-shape)
72˚ -> -144˚, (a |-> aaa) (anti-clockwise -> (anti)butterfly-shape)
72˚ -> -72˚
~~~
so,
~~~
|Int(D5)| = |Aut(D5)| / 2
~~~
why not compare D3, D5, D7 ?
~~~
1 * |Int(D3)| = |Aut(D3)|
2 * |Int(D5)| = |Aut(D5)|
3 * |Int(D7)| = |Aut(D7)|
~~~
more generally,
~~~
Aut(Dn) = n * φ(n)
where φ(n) denotes Euler's number
~~~
# Galois
The difference between $ D3 $ and $ D5 $ is
extended to $ A _ 5 $ $ \sim $ $ G( T _ {20} ) $ $ \sim $ $ G( T _ {12} ) $
As $ Aut( D _ 5 ) $ contains Butterfly-shape,
$ Aut( A _ 5 ) $ contains more complex-shape that is not included in A5 itself.
So the structure of $ Aut( A _ 5 ) $ cause that we cannot solve algebraic quintic equation.
# Haskell code
~~~ Haskell
-- Type Definition
data Mat a = Mat [[a]]
-- Show Intance of Matrix
showRow [] = ""
showRow (x:xs) = show x ++ "\t\b" ++ showRow xs
instance (Show a) => Show (Mat a) where
show (Mat []) = ""
show (Mat (x:xs)) = showRow x ++ "\n" ++ show (Mat xs)
-- Define Lengths
len_row x = maximum $ map length x
len_col x = length x
-- Define Operators
instance (Num a) => Num (Mat a) where
(Mat a) + (Mat b) =
let m = max(len_row a)(len_row b) in
let n = max(len_col a)(len_col b) in
Mat [
[ (a!!i!!j) + (b!!i!!j) | j <- [0..(n-1)]]
| i <- [0..(m-1)]
]
(Mat a) - (Mat b) =
let m = max(len_row a)(len_row b) in
let n = max(len_col a)(len_col b) in
Mat [
[ (a!!i!!j) - (b!!i!!j) | j <- [0..(n-1)]]
| i <- [0..(m-1)]
]
(Mat a) * (Mat b) =
let l = len_row a - 1 in
let m = len_col a - 1 in
let n = len_col b - 1 in
Mat [
[ sum [((a!!i!!k)*(b!!k!!j)) | k<-[0..m]] | j<-[0..n] ]
| i<-[0..l]
]
-------------------- an expample usage --------------------
-- Now we think of
-- * Symmetry Group S_4
-- * Alternation Group A_4
-- A_4 is generated just by a & b
-- S_4 is generated just by a, b & z
-- the Matrix is like
-- [0 0 1]
-- [y z x] = [x y z][1 0 0]
-- [0 1 0]
-- here x,y or z axis implies each axis of 3 180˚rotations
-----------------------------------------------------------
e = aaa
a = Mat [
[ 0, 0, 1],
[ 1, 0, 0],
[ 0, 1, 0]]
b = Mat [
[ 0, 0, 1],
[-1, 0, 0],
[ 0,-1, 0]]
c = b*b*a*a
d = a*a*b*b
aa = a*a
bb = b*b
cc = c*c
dd = d*d
h = b*a*a
i = a*b*a
j = a*a*b
z = Mat [
[ 0, 0,-1],
[ 0,-1, 0],
[-1, 0, 0]]
za = z * a
zb = z * b
zc = z * c
zd = z * d
zaa = z * aa
zbb = z * bb
zcc = z * cc
zdd = z * dd
zh = z * h
zi = z * i
zj = z * j
~~~