The Outer Automorphism of S6

21 Pages·2016·0.2 MB·English

The Outer Automorphism of S
6
Meena Jagadeesan1 Karthik Karnik2
Mentor: Akhil Mathew
1PhillipsExeterAcademy
2MassachusettsAcademyofMathandScience
PRIMES Conference, May 2016
MeenaJagadeesan,KarthikKarnikMentor:AkhilMathew PRIMESConference,May2016
TheOuterAutomorphismofS6
What is a Group?
Definition
A group G is a set of elements together with an operation that
satisfies the four fundamental properties: closure, associativity,
identity, and inverses.
Our Focus: Symmetric group S
n
Set = permutations π of n elements; Operation = ◦ (composition)
Closure: For π ,π ∈ S , π ◦π ∈ S
1 2 n 1 2 n
Associativity: For π ,π ,π ∈ S ,
1 2 3 n
(π ◦π )◦π = π ◦(π ◦π )
1 2 3 1 2 3
Identity: Take permutation e = π such that π(i) = i for all
1 ≤ i ≤ n.
Inverses: For inverse of π , take π−1 such that π−1(i) = j iff
1 1 1
π (j) = i.
1
MeenaJagadeesan,KarthikKarnikMentor:AkhilMathew PRIMESConference,May2016
TheOuterAutomorphismofS6
What is an Automorphism Group?
Definition
Given a group G, the automorphism group Aut(G) is the group
consisting of all isomorphisms from G to G (bijective mappings
that preserve the structure of G.)
(cid:8)(cid:0) (cid:1) (cid:0) (cid:1)(cid:9)
Example: Aut(S ) (S = 1 2 , 1 2 )
2 2 1 2 2 1
(cid:0) (cid:1)
1 2 is the identity
1 2
(cid:0) (cid:1)
An automorphism f must send 1 2 to itself
1 2
(cid:0) (cid:1)
f must send 1 2 to itself
2 1
f is the identity e
This means Aut(S ) is trivial
2
MeenaJagadeesan,KarthikKarnikMentor:AkhilMathew PRIMESConference,May2016
TheOuterAutomorphismofS6
What is an Automorphism Group?
Example: Aut(Z ×Z ) (Z ×Z is group of pairs of elements
2 2 2 2
modulo 2)
Observe that Z ×Z consists of the identity and three other
2 2
elements of order 2
An automorphism f must send the identity to itself
The other three elements are permuted by f
This means that Aut(Z ×Z ) ∼= S
2 2 3
MeenaJagadeesan,KarthikKarnikMentor:AkhilMathew PRIMESConference,May2016
TheOuterAutomorphismofS6
What is an Inner Automorphism?
Definition
An inner automorphism is an automorphism of the form
f : x (cid:55)→ a−1xa for a fixed a ∈ G.
This gives us an automorphism of G for each element of G.
There might be repeats. (If G is commutative, then f is always
trivial)
Trivial inner automorphisms = Z(G) (elements that commute with
all other elements)
Theorem
The inner automorphisms Inn(G) form a normal subgroup of
∼
Aut(G) and Inn(G) = G/Z(G)
MeenaJagadeesan,KarthikKarnikMentor:AkhilMathew PRIMESConference,May2016
TheOuterAutomorphismofS6
Complete Groups
Definition
A group G is complete if G is centerless (no nontrivial center)
and every automorphism of G is an inner automorphism.
∼
G complete ⇒ Aut(G) = G
Theorem
S is complete for n (cid:54)= 2,6.
n
The center of S is itself. S is a genuine exception:
2 6
Theorem (H¨older)
There exists exactly one outer automorphism of S (up to
6
composition with an inner automorphism), so that
|Aut(S )| = 1440.
6
MeenaJagadeesan,KarthikKarnikMentor:AkhilMathew PRIMESConference,May2016
TheOuterAutomorphismofS6
Transpositions
Definition
A transposition is a permutation π ∈ S that fixes exactly n−2
n
elements (and flips the remaining two elements).
Lemma
An automorphism of S preserves transpositions if and only if it is
n
an inner automorphism.
MeenaJagadeesan,KarthikKarnikMentor:AkhilMathew PRIMESConference,May2016
TheOuterAutomorphismofS6
Proof Overview: Completeness of S for n (cid:54)= 2,6
n
Let T be the conjugacy class in S consisting of products of
k n
k disjoint transpositions.
A permutation π is an involution if and only if it lies in some
T .
k
If f ∈Aut(S ), then f(T )=T for some k.
n 1 k
It suffices to show |T | =(cid:54) |T | for k (cid:54)= 1.
k 1
This is true for n(cid:54)=6.
For n=6, it turns out that |T |=|T | is the only exception.
1 3
MeenaJagadeesan,KarthikKarnikMentor:AkhilMathew PRIMESConference,May2016
TheOuterAutomorphismofS6
Transitive Group Actions
Definition
A group G acts on a set X if each element g is a permutation π
g
of the set X satisfying π = e and π = π ◦π .
e gf g f
Example 1: S acts on {1,2,3,...,n}.
n
Example 2: S acts on {1,2,3,...,n} by fixing n.
n−1
Example 3: G acts on the coset space G/H by multiplication.
Definition
A group action is transitive if for each pair (x,y) ∈ X2 there
exists g ∈ G such that g(x) = y.
Examples 1 and 3 are transitive group actions, but Example 2 is
not.
MeenaJagadeesan,KarthikKarnikMentor:AkhilMathew PRIMESConference,May2016
TheOuterAutomorphismofS6
Proof Overview
(Existence of an Outer Automorphism of S )
6
Key Step: Construct a 120-element subgroup H of S that acts
6
transitively on {1,2,3,4,5,6}.
This subgroup cannot be S or any of its conjugates (none are
5
transitive subgroups).
Consider the action of S on the 6-element coset space S /H.
6 6
Let f the corresponding mapping from S to S .
6 6
Note that f : H (cid:55)→ S . (H fixes coset consisting of H and
5
permutes all other cosets. H has order 120, the same as S .)
5
H (the preimage of S in f) is transitive.
5
The preimage of S is not conjugate to S .
5 5
f cannot be inner.
MeenaJagadeesan,KarthikKarnikMentor:AkhilMathew PRIMESConference,May2016
TheOuterAutomorphismofS6