Lecture 13 The alternating group and more on group actions · Reviewing groups actions Recall that...

Lecture 13The alternating group and more on group actions

Reviewing groups actionsRecall that a group G has a group action on a set A

σg : A→ A where σg : a 7→ g · a

if1 · a = a and g · (h · a) = (gh) · a

for all g, h ∈ G, a ∈ A.

The homomorphism

ϕ : G→ SA defined by ϕ : g → σg

is called the permutation representation induced (or afforded) bythe action .

Proposition

For any group G and any non-empty set A, there is a bijectionbetween

{ group actions of G on A } ↔ { homomorphisms ϕ : G→ SA}.

(We showed this bijection in Lecture 5)

if1 · a = a and g · (h · a) = (gh) · a

for all g, h ∈ G, a ∈ A. The homomorphism

Proposition

if1 · a = a and g · (h · a) = (gh) · a

for all g, h ∈ G, a ∈ A. The homomorphism

Proposition

(∗) The stabilizer of any a ∈ A is

Ga = {g ∈ G | g · a = a},

and is a subgroup of G.

(∗) The kernel of the action is

ker = {g ∈ G | g · a = a for all a ∈ A} =⋂a∈A

and is a subgroup of G.(∗) An action is faithful if and only if the kernel is trivial.(∗) The relation on A defined by

a ∼ b if and only if a = g · b for some g ∈ G

is an equivalence relation.

Proposition

For each a ∈ A, the number of elements in the equivalence classcontaining a is |G : Ga|.

Ga = {g ∈ G | g · a = a},

and is a subgroup of G.(∗) The kernel of the action is

and is a subgroup of G.

(∗) An action is faithful if and only if the kernel is trivial.(∗) The relation on A defined by

Proposition

Ga = {g ∈ G | g · a = a},

and is a subgroup of G.(∗) An action is faithful if and only if the kernel is trivial.

(∗) The relation on A defined by

Proposition

Ga = {g ∈ G | g · a = a},

Proposition

Ga = {g ∈ G | g · a = a},

Proposition

DefinitionLet G be a group acting on a set A.

1. The equivalence class Oa = {g · a | g ∈ G} is called the orbitof a.

2. The action of G on A is called transitive if there is only oneorbit.

Back to the symmetric group

The symmetric group Sn is the group of permutations of n objects.

We can write a permutation as the product of disjoint cycles:

σ = (1324)(57)

Advantage: unique up to reordering cycles (we still have to provethis!).Disadvantage: How do we write down a presentation of Sn?

σ = (1324)(57)

Reconstructing the symmetric groupDefinition: The 2-cycles in Sn are called transpositions.

(a1am) · · · (a1a4)(a1a3)(a1a2) =

(a1a2a3a4 . . . am),

every cycle can be written as the product of transpositions, and soevery permutation can be written as the product of transpositions.

Caution: not unique!For example

(1324)(57) = (14)(12)(13)(57)

= (23)(12)(24)(57)

= (23)(12)(34)(23)(34)(57)

= (23)(12)(34)(23)(34)(56)(67)(56)

Sn = 〈T 〉 where T = {(i j) | 1 ≤ i < j ≤ n}.

(a1am) · · · (a1a4)(a1a3)(a1a2) = (a1a2a3a4 . . . am),

(1324)(57) = (14)(12)(13)(57)

= (23)(12)(24)(57)

= (23)(12)(34)(23)(34)(57)

= (23)(12)(34)(23)(34)(56)(67)(56)

Sn = 〈T 〉 where T = {(i j) | 1 ≤ i < j ≤ n}.

(1324)(57) = (14)(12)(13)(57)

= (23)(12)(24)(57)

= (23)(12)(34)(23)(34)(57)

= (23)(12)(34)(23)(34)(56)(67)(56)

Sn = 〈T 〉 where T = {(i j) | 1 ≤ i < j ≤ n}.

(1324)(57) = (14)(12)(13)(57)

= (23)(12)(24)(57)

= (23)(12)(34)(23)(34)(57)

= (23)(12)(34)(23)(34)(56)(67)(56)

Sn = 〈T 〉 where T = {(i j) | 1 ≤ i < j ≤ n}.

What is unique?

Sn = 〈T 〉 where T = {(i j) | 1 ≤ i < j ≤ n}.

ClaimFix σ ∈ Sn. Every expression of σ as a product of transpositionshas length of the same parity.

Proof by way of the action on polynomials:Recall that Sn acts on Z[x1, . . . , xn] by

σ · p(x1, . . . , xn) = p(xσ(1), . . . , xσ(n)).

∆ =∏

1≤i<j≤n(xi − xj) = (x1 − x2)(x1 − x3) · · · (xn−1 − xn)︸︷︷︸

n(n−1)/2 terms

What is unique?

Sn = 〈T 〉 where T = {(i j) | 1 ≤ i < j ≤ n}.

ClaimFix σ ∈ Sn. Every expression of σ as a product of transpositionshas length of the same parity.

Proof by way of the action on polynomials:Recall that Sn acts on Z[x1, . . . , xn] by

σ · p(x1, . . . , xn) = p(xσ(1), . . . , xσ(n)).

∆ =∏

1≤i<j≤n(xi − xj) = (x1 − x2)(x1 − x3) · · · (xn−1 − xn)︸︷︷︸

n(n−1)/2 terms

Some examples

∆3 = (x1 − x2) (x1 − x3) (x2 − x3)

∆4 = (x1 − x2) (x1 − x3) (x1 − x4) (x2 − x3) (x2 − x4) (x3 − x4)

∆5 = (x1 − x2)(x1 − x3)(x1 − x4)(x1 − x5)(x2 − x3)(x2 − x4)(x2 − x5)(x3 − x4)(x3 − x5)(x4 − x5)

Consider

σ ·∆ =∏

1≤i<j≤n(xσ(i) − xσ(j))

= (−1)#{1≤i<j≤n | σ(j)<σ(i)}∆.

For example, if n = 4 and σ = (1324), then

(−1)5

(x1 − x2)

(x1 − x3)

(x1 − x4)

(x2 − x3)

(x2 − x4)

(x3 − x4)

{+1 if σ ·∆ = ∆,

−1 if σ ·∆ = −∆.

Then ε(σ) is the sign of σ.

Proposition

The map ε : Sn → {±1}× is a homomorphism.

Proposition

All transpositions have negative sign, i.e. ε((i j)) = −1. Therefore,

(1) the homomorphism ε is surjective for all n > 1, and

(2) for any expression of σ ∈ Sn as the product of transpositions,the parity of the length of that product is determined; namelythe length is

even if ε(σ) = 1, and

odd if ε(σ) = −1.

{+1 if σ ·∆ = ∆,

−1 if σ ·∆ = −∆.

Proposition

{+1 if σ ·∆ = ∆,

−1 if σ ·∆ = −∆.

Proposition

DefinitionWe call a permutation even if ε(σ) = 1 and odd if ε(σ) = −1.

The alternating group An is the kernel of ε : Sn → {±1}×, i.e.

An = {σ ∈ Sn | σ is even }.

Caution: even/oddAn m-cycle is even if m is odd, and is odd if m is even.

Examples

A1 = {1}

A2 = {1}

A3 = {1} ∪ {3-cycles} = {1, (123), (132)}

A4 = {1} ∪ {3-cycles} ∪ { two disjoint 2-cycles }= {1, (123), (124), (132), (134), (142),

(143), (234), (243), (12)(34), (13)(24), (14)(23)}

Lecture 13 The alternating group and more on group actions · Reviewing groups actions Recall that...

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Transcript of Lecture 13 The alternating group and more on group actions · Reviewing groups actions Recall that...