Sporadic Symmetry - (The Remarkable Behaviour of the ...sh15083/talks/2015.04.17_mt5999.pdf · a simple group in one of the 16 families of groups of Lie type; one of the 26 sporadic

Sporadic Symmetry(The Remarkable Behaviour of the Mathieu Groups)

Scott Harper

MT5999 Presentation

17th April 2015

Scott Harper Sporadic Symmetry 17th April 2015 1 / 9

Sporadic Symmetry(The Remarkable Behaviour of the Mathieu Groups)

Scott Harper

MT5999 Presentation

17th April 2015


It is a remarkable fact that ...

From Sphere Packings, Lattices and Groups by Conway and Sloane,

“At one point while working on this book we evenconsidered adopting a special abbreviation for ‘It is aremarkable fact that’ since this phrase seemed to occurso often. But in fact we have tried to avoid such phrasesand to maintain a scholarly decorum of language.”

Theorem

The symmetric group Sn has a non-trivial outer automorphism ifand only if n = 6.


It is a remarkable fact that ...

From Sphere Packings, Lattices and Groups by Conway and Sloane,

“At one point while working on this book we evenconsidered adopting a special abbreviation for ‘It is aremarkable fact that’ since this phrase seemed to occurso often. But in fact we have tried to avoid such phrasesand to maintain a scholarly decorum of language.”

Theorem

The symmetric group Sn has a non-trivial outer automorphism ifand only if n = 6.


Transitivity

Definition

Let G be a group acting on a set X . We sayG acts

transitively if for all x , y ∈ X thereexists g ∈ G such that xg = y ;

k-transitively if for all sequences ofdistinct points(x1, . . . , xk), (y1, . . . , yk) ∈ X k thereexists g ∈ G such that xig = yi ;

sharply k-transitively if the above g isthe unique such element.

4

1

3

2


Transitivity

Definition





4

1

3

2


Transitivity

Definition





4

1

3

2


Transitivity

Definition





4

1

3

2


Transitivity

Definition





4

1

3

2


Simplicity

Theorem (The Classification of Finite Simple Groups)

Every finite simple group is isomorphic to one of the following groups:

a cyclic group of prime order;

an alternating group of degree at least 5;

a simple group in one of the 16 families of groups of Lie type;

one of the 26 sporadic simple groups.

Theorem

Let G be a group acting k-transitively on a set. If G is not a symmetric oralternating group then k ≤ 5.

Moreover,

if k = 5 then G is the Mathieu group M12 or M24;

if k = 4 then G is the Mathieu group M11 or M23.

Note: M22 acts 3-transitively.


Simplicity







Theorem


Moreover,





Simplicity







Theorem


Moreover,





Simplicity







Theorem

Let G be a group acting k-transitively on a set. If G is not a symmetric oralternating group then k ≤ 5. Moreover,





Simplicity







Theorem

Let G be a group acting k-transitively on a set. If G is not a symmetric oralternating group then k ≤ 5. Moreover,





Steiner Systems

The affine plane AG2(3)

Definition

An S(t, k , v) Steiner system is aset X of v points and a set ofk-element subsets of X such thatany t points lie in a unique suchsubset.

Example

The affine plane AG2(3) is anS(2, 3, 9) Steiner system.


Steiner Systems


Definition


Example



Steiner Systems


Definition


Example



Witt Geometries and Mathieu Groups

S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x ≤ Aut(W11) =: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3)

(M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)



S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x ≤ Aut(W11) =: M11

S(3, 4, 10)

W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3)


= =

H ≤ AGL2(3)



S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11)

W11 (M12)x ≤ Aut(W11) =: M11

S(3, 4, 10)

W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3)


= =

H ≤ AGL2(3)



S(5, 6, 12)

W12 M12 := Aut(W12)

S(4, 5, 11)

W11 (M12)x ≤ Aut(W11) =: M11

S(3, 4, 10)

W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3)


= =

H ≤ AGL2(3)



S(5, 6, 12) W12

M12 := Aut(W12)

S(4, 5, 11) W11

(M12)x ≤ Aut(W11) =: M11

S(3, 4, 10) W10

(M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3)


= =

H ≤ AGL2(3)



S(5, 6, 12) W12

M12 := Aut(W12)

S(4, 5, 11) W11

(M12)x ≤ Aut(W11) =: M11

S(3, 4, 10) W10

(M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3)

(M12)xyz ≤

Aut(AG2(3))

=

=

H ≤

AGL2(3)



S(5, 6, 12) W12

M12 := Aut(W12)

S(4, 5, 11) W11

(M12)x ≤ Aut(W11) =: M11

S(3, 4, 10) W10

(M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3)

(M12)xyz ≤

Aut(AG2(3))

=

=

H ≤ AGL2(3)



S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11

(M12)x ≤

Aut(W11)

=: M11

S(3, 4, 10) W10

(M12)xy ≤

Aut(W10)

S(2, 3, 9) AG2(3)

(M12)xyz ≤

Aut(AG2(3))

=

=

H ≤ AGL2(3)



S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x ≤ Aut(W11)

=: M11

S(3, 4, 10) W10

(M12)xy ≤

Aut(W10)

S(2, 3, 9) AG2(3)

(M12)xyz ≤

Aut(AG2(3))

=

=

H ≤ AGL2(3)



S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x ≤ Aut(W11)

=: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3)

(M12)xyz ≤

Aut(AG2(3))

=

=

H ≤ AGL2(3)



S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x ≤ Aut(W11)

=: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

=

=

H ≤ AGL2(3)



S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x ≤ Aut(W11)

=: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)



S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x ≤ Aut(W11)

=: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)

Theorem

Suppose that

- G acts transitively on X

- Gx acts k-transitively on X \ {x}, for some x ∈ X .

Then G acts (k + 1)-transitively on X .



S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x ≤ Aut(W11)

=: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)

Theorem

Suppose that


- Gx acts sharply k-transitively on X \ {x}, for some x ∈ X .

Then G acts sharply (k + 1)-transitively on X .



S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x ≤ Aut(W11)

=: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)

Theorem

Suppose that






S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x ≤ Aut(W11)

=: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)

Theorem

Suppose that






S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x ≤ Aut(W11)

=: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)

Theorem

Suppose that






S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x = Aut(W11)

=: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)

Theorem

Suppose that






S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x = Aut(W11) =: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)

Theorem

Suppose that






S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x = Aut(W11) =: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)

Theorem

Suppose that






S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x = Aut(W11) =: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)

Theorem

Suppose that

- G acts k-transitively on X for k ≥ 3 and |X | not equal to 3 or 2n

- Gx is simple, for some x ∈ X .

Then G is simple.



S(5, 6, 12) W12 M12 := Aut(W12)

S(4, 5, 11) W11 (M12)x = Aut(W11) =: M11

S(3, 4, 10) W10 (M12)xy ≤ Aut(W10)

S(2, 3, 9) AG2(3) (M12)xyz ≤ Aut(AG2(3))

= =

H ≤ AGL2(3)

Theorem

Suppose that

- G acts k-transitively on X for k ≥ 3 and |X | not equal to 3 or 2n

- Gx is simple, for some x ∈ X .

Then G is simple.


M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6 SB SC S6

G

The map φ is a homomorphism, aninjection and a surjection.

Fact: φ−1ψ : SB → SC is anisomorphism.

Example: φ−1ψ : (b1 b2) 7→(c1 c2)(c3 c4)(c5 c6)

Fact: βφ−1ψγ : S6 → S6 isan isomorphism.

Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)

So βφ−1ψγ is a non-trivialouter automorphism of S6.


M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6 SB SC S6

G





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B

C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6 SB SC S6

G





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6 SB SC S6

G





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6 SB SC S6

G





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B

ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6

SB

SC S6

Gφ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B

ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6

SB

SC S6

Gφ

The map φ is a homomorphism,

aninjection and a surjection.




Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B

ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6

SB

SC S6

Gφ

The map φ is a homomorphism, aninjection

and a surjection.




Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B

ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6

SB

SC S6

Gφ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6

SB SC

S6

Gφ ψ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6

SB SC

S6

Gφ ψ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6

SB SC

S6

Gφ ψ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6

SB SC

S6

Gφ ψ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6

SB SC

S6

Gφ ψ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6 SB SC

S6

G

β

φ ψ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6 SB SC S6

G

β γ

φ ψ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6 SB SC S6

G

β γ

φ ψ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6 SB SC S6

G

β γ

φ ψ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



M12 acting on W12

φ : g 7→ g |B ψ : g 7→ g |C

B C

123456

b1b2b3b4b5b6

c1c2c3c4c5c6

123456

S6 SB SC S6

G

β γ

φ ψ





Example: φ−1ψ : (1 2) 7→(1 2)(3 4)(5 6)



Mathieu Groups

“These apparently sporadic simple groups would probably repay a closerexamination than they have yet received.”

(William Burnside)

Any questions?


Mathieu Groups

“These apparently sporadic simple groups would probably repay a closerexamination than they have yet received.”

(William Burnside)

Any questions?


Documents

Sporadic Symmetry - (The Remarkable Behaviour of the ...sh15083/talks/2015.04.17_mt5999.pdf · a simple group in one of the 16 families of groups of Lie type; one of the 26 sporadic