Automorphisms of finite groups G a finite group. : G Aut ( V ) a - - PowerPoint PPT Presentation

A construction for the outer automorphism of S6

Padraig Ó Catháin joint work with Neil Gillespie and Cheryl Praeger

University of Queensland

5 August 2013

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

Automorphisms of finite groups

G a finite group. ρ : G → Aut(V) a representation. Let σ ∈ Aut(G). When can σ be realised as an element of Aut(V)? I.e. when does there exist S ∈ Aut(V) such that, for all g ∈ G: ρ(xσ) = S−1ρ(x)S (Obvious) sufficient condition: σ is inner. (Obvious) necessary condition: χ(xσ) = χ(x) for all x ∈ G.

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

Motivation: M12

M12 has two conjugacy classes of subgroups M11. Coset action on either class is 5-transitive on 12 points. The outer automorphism of M12 swaps classes of M11s, and hence the two actions. These actions cannot be (linearly) equivalent: the traces are different, e.g. P = 1 2 3 4 5 6 7 8 9 10 11 12 5 2 3 1 9 10 6 7 4 8 11 12

• Pσ

=

• 1

2 3 4 5 6 7 8 9 10 11 12 10 1 6 2 7 5 3 9 8 4 12 11

• Padraig Ó Catháin

Outer automorphism of S6 5 August 2013

Motivation: M12

Theorem (M. Hall, 1962) Let H be a Hadamard matrix of order 12. Modulo the centre of Aut(H), the automorphisms of H are the Mathieu group M12 of order 12 · 11 · 10 · 9 · 8 = 95040. Here M12 is represented as a quintuply transitive group of monomial permutations on the columns or rows of H. The row and column representations of H are isomorphic, but the correspondence given by P = HQH−1 determines an outer automorphism of M12 of order 2. (Conway and Elkies also note that (P, Q) → (Q, P) exhibits the outer automorphism of M12.)

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

Let H be Hadamard of order 12. Aut(H) = {(P, Q) | PHQ⊤ = H, where P, Q are ± 1-monomial} Consider the representations α : 2.M12 → P and β : 2.M12 → Q: α(x) = Hβ(x)H−1. By Hall: β(x) = α(xσ) for some outer automorphism σ of M12. So for every x ∈ 2.M12, we have α(xσ) = H−1α(x)H, a linear representation of the outer automorphism of 2.M12.

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

Construction for the outer automorphism of M12

While the outer automorphism of M12 (acting on 12 points) cannot be realised linearly, it almost can. Given an element of M12,

1

lift to 2.M12

2

conjugate by H

3

project back onto M12. Sample automorphism: π−1(P) =

• 1

2 3 4 5 6 7 8 9 10 11 12 −5 2

• 3

−1 −9 10 6 7 −4 8 11

• 12
• π−1(Pσ)

=

• 1

2 3 4 5 6 7 8 9 10 11 12 10 1 6 2 7 5 3 9 8 4 12 11

• =

π−1(P)H

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

The outer automorphism of S6

Possibly the most famous outer automorphism of any group: (1, 2) → (1, 2)(3, 6)(4, 5) (1, 2, 3) → (1, 5, 6)(2, 3, 4) (1, 2, 3, 4, 5, 6) → (1, 5)(2, 3, 6) etc.

1

Described by Sylvester: duads, synthemes, totals, etc.

2

Exhibited by the actions of Aut(K6) on points and a set of

• ne-factors

3

Exhibited by action of S6 on certain 2-colourings of K5

4

Via the isomorphism S6 ∼ = Sp4(2)

5

As a subgroup of M12, etc. Can we find a matrix H which intertwines (lifts of) these representations of S6?

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

A fragment from Moorhouse

Moorhouse: classification of the complex Hadamard matrices with doubly transitive automorphism groups. A sporadic example: H =         1 1 1 1 1 1 1 1 ω ω ω ω 1 ω 1 ω ω ω 1 ω ω 1 ω ω 1 ω ω ω 1 ω 1 ω ω ω ω 1         which has automorphism group 3.A6. Note: HH† = 6I6.

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

Aside: what we were actually doing...

Attempting to construct codes with interesting automorphism groups from the rows of Hadamard matrices. The rows of the matrix (considered over F4) generate ‘the hexacode’, used by Conway and Sloane to describe the Golay code. Considered over F3, we obtained a new nonlinear uniformly packed code, and a new frequency permutation array, etc. These ideas should extend to larger families of Hadamard matrices...

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

H intertwines two representations of 3.A6. Restricting to A6 we get inequivalent representations of A6. So every automorphism of H is of the form ρ(x)Hρ(xσ). Now, ρ extends to a representation of 3.S6. If we take y to be an

• dd permutation, then ρ(y)Hρ(yσ) = H†. But H is not Hermitian...

In any case, H could never intertwine representations of 3.S6: involutions of shape 142 have non-zero trace over C, but are mapped to elements of shape 23. (Cf. elements of order 3.)

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

The split-quaternions, Ξ

Discovered by Hamilton Cockle in 1849 4-dimensional over R: generated by {1, i, τ, iτ}, where i2 = −1 τ 2 = 1 τiτ = −i. Isomorphic to M2(R): i → −1 1

• τ →

1 −1

• Not a division algebra: (i + τi)2 = 0 (Cockle’s "impossibles").

Contains C ∼ = 1, i as a subalgebra.

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

Take S = τI6, then, for any odd permutation y ∈ S6, we have: Sρ(y)Hρ(yσ)S = SH†S = H So odd permutations can be made into Ξ-linear automorphisms of

• H. Note that τ and τi have trace 0: so all odd permutations lift to

elements of equal trace. Even permutations are defined over C, odd permutations involve τ. We have 3.S6 acting on H with the row and column actions differing by an outer automorphism of S6. To compute the image of σ ∈ S6 under an outer automorphism: lift σ to 3.S6, if odd, multiply by S, conjugate by H, restrict to S6.

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

Conclusion

A construction of the outer automorphism for S6 which does not depend on finding two non-conjugate subgroups of index 6. Could be realised as a C-linear 12-dimensional representation. Question: Which automorphisms of which representations of which groups can be realised linearly in this sense? Question: Can we find other interesting combinatorial objects hiding in the rings of intertwiners of representations?

Padraig Ó Catháin Outer automorphism of S6 5 August 2013

Thank you!

Padraig Ó Catháin Outer automorphism of S6 5 August 2013