Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math...

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Properties of Homomorphisms Recall: A function φ : G ¯ G is a homomorphism if φ(ab)= φ(a)φ(b)a, b G . Let φ : G ¯ G be a homomorphism, let g G , and let H G . Properties of elements Properties of subgroups 1. φ(e G )= e ¯ G 1. φ(H ) ¯ G . 2. φ(g n )=(φ(g )) n n Z. 2. H cyclic = φ(H ) cyclic. 3. If |g | is finite, |φ(g )| |g |. 3. H Abelian = φ(H ) Abelian. 7. ¯ K ¯ G = φ -1 ( ¯ K ) G . 4. H / G = φ(H ) (G ) 8. ¯ K / ¯ G = φ -1 ( ¯ K ) / G . Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 1 / 12

Transcript of Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math...

Page 1: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Properties of Homomorphisms

Recall: A function φ : G → G is a homomorphism if

φ(ab) = φ(a)φ(b)∀a, b ∈ G .

Let φ : G → G be a homomorphism, let g ∈ G , and let H ≤ G .

Properties of elements Properties of subgroups

1. φ(eG ) = eG 1. φ(H) ≤ G .

2. φ(gn) = (φ(g))n∀n ∈ Z. 2. H cyclic =⇒ φ(H) cyclic.

3. If |g | is finite, |φ(g)|∣∣ |g |. 3. H Abelian =⇒ φ(H) Abelian.

7. K ≤ G =⇒ φ−1(K ) ≤ G .

4. H / G =⇒ φ(H) / φ(G )

8. K / G =⇒ φ−1(K ) / G .

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 1 / 12

Page 2: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Properties of Homomorphisms

Let φ : G → G be a homomorphism, let g ∈ G , and let H ≤ G .

Properties of elements Properties of subgroups

1. φ(eG ) = eG 1. φ(H) ≤ G .

2. φ(gn) = (φ(g))n for all n ∈ Z. 2. H cyclic =⇒ φ(H) cyclic.

3. If |g | is finite, |φ(g)| divides |g |. 3. H Abelian =⇒ φ(H) Abelian.

4. Ker(φ) ≤ G 4. H / G =⇒ φ(H) / φ(G )

7. K ≤ G =⇒ φ−1(K ) ≤ G .

8. K / G =⇒ φ−1(K ) / G .

Remember thatKer(φ)

def= {g ∈ G |φ(g) = idG}.

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 2 / 12

Page 3: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

In Class Work

1. Find the kernel of the homomorphism p : G ⊕H → G by p(g , h) = g .

2. Find the kernel of the homomorphism i : H → G ⊕ H byi(h) = (eG , h).

3. Let G be a group of permutations. For each σ ∈ G , define

sgn(σ) =

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Prove that sgn is a homomorphism from G to the multiplicativegroup {+1,−1}. What is the kernel?

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 3 / 12

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Solutions:

1. Find the kernel of the homomorphism p : G ⊕H → G by p(g , h) = g .

Ker(p)def= {(g , h) ∈ G ⊕ H

∣∣p(g , h) = eG}= {(g , h)

∣∣g = eG}= {(eG , h)

∣∣h ∈ H}= {eG} ⊕ H.

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 4 / 12

Page 5: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions:2. Find the kernel of the homomorphism i : H → G ⊕ H by

i(h) = (eG , h).

Ker(i)def= {h ∈ H

∣∣i(h) = eG⊕H = (eG , eH)}= {h ∈ H

∣∣(eG , h) = (eG , eH)}= {eH}.

Or...

Since i is a homomorphism, i(eH) = eG⊕H .

Since we showed Wednesday that i is 1-1, nothing besides the identity canmap to the identity. Thus the kernel, which is the set of all things thatmap to the identity, contains only the identity.

Notice: This shows that the kernel of any 1-1 homomorphism consists onlyof the identity!

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 5 / 12

Page 6: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions:

3. Let G be a group of permutations. For each σ ∈ G , define

sgn(σ) =

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Prove that sgn is a homomorphism from G to the multiplicativegroup {+1,−1}. What is the kernel?

To show that sgn is a homomorphism, NTS sgn is a well-defined functionand is operation-preserving.

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 6 / 12

Page 7: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions:

3. Let G be a group of permutations. For each σ ∈ G , define

sgn(σ) =

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Prove that sgn is a homomorphism from G to the multiplicativegroup {+1,−1}. What is the kernel?

Is sgn well-defined?

Suppose that σ1 = σ2.

Then since every permutation’s factorization into transpositions will beeither always odd or always even, either both σ1 and σ2 are even or bothσ1 and σ2 are odd.

Thus sgn(σ1) = sgn(σ2), and so sgn is a well-defined function.

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 6 / 12

Page 8: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions to 3, continued

Recall: sgn(σ)def=

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Is sgn operation-preserving?

sgn(αβ) =

{+1 if αβ is even

αβ is even ⇔ α, β

−1 if αβ is odd

both even or both odd

sgn(αβ) =

{+1 if α, β both even or both odd

−1 if one is even, the other odd

=

{+1 if sgn(α) = sgn(β)

i.e. if sgn(α) = sgn(β) = ±1

−1 if sgn(α) 6= sgn(β)

i.e. if sgn(α) = ±1, sgn(β) = ∓1

=

{+1 if sgn(α)sgn(β) = +1

−1 if sgn(α)sgn(β) = −1

= sgn(α)sgn(β) Thus sgn preserves the group operation

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 7 / 12

Page 9: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions to 3, continued

Recall: sgn(σ)def=

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Is sgn operation-preserving?

sgn(αβ) =

{+1 if αβ is even αβ is even ⇔ α, β

−1 if αβ is odd both even or both odd

sgn(αβ) =

{+1 if α, β both even or both odd

−1 if one is even, the other odd

=

{+1 if sgn(α) = sgn(β)

i.e. if sgn(α) = sgn(β) = ±1

−1 if sgn(α) 6= sgn(β)

i.e. if sgn(α) = ±1, sgn(β) = ∓1

=

{+1 if sgn(α)sgn(β) = +1

−1 if sgn(α)sgn(β) = −1

= sgn(α)sgn(β) Thus sgn preserves the group operation

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 7 / 12

Page 10: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions to 3, continued

Recall: sgn(σ)def=

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Is sgn operation-preserving?

sgn(αβ) =

{+1 if αβ is even αβ is even ⇔ α, β

−1 if αβ is odd both even or both odd

sgn(αβ) =

{+1 if α, β both even or both odd

−1 if one is even, the other odd

=

{+1 if sgn(α) = sgn(β)

i.e. if sgn(α) = sgn(β) = ±1

−1 if sgn(α) 6= sgn(β)

i.e. if sgn(α) = ±1, sgn(β) = ∓1

=

{+1 if sgn(α)sgn(β) = +1

−1 if sgn(α)sgn(β) = −1

= sgn(α)sgn(β) Thus sgn preserves the group operation

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 7 / 12

Page 11: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions to 3, continued

Recall: sgn(σ)def=

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Is sgn operation-preserving?

sgn(αβ) =

{+1 if αβ is even αβ is even ⇔ α, β

−1 if αβ is odd both even or both odd

sgn(αβ) =

{+1 if α, β both even or both odd

−1 if one is even, the other odd

=

{+1 if sgn(α) = sgn(β)

i.e. if sgn(α) = sgn(β) = ±1

−1 if sgn(α) 6= sgn(β)

i.e. if sgn(α) = ±1, sgn(β) = ∓1

=

{+1 if sgn(α)sgn(β) = +1

−1 if sgn(α)sgn(β) = −1

= sgn(α)sgn(β) Thus sgn preserves the group operation

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 7 / 12

Page 12: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions to 3, continued

Recall: sgn(σ)def=

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Is sgn operation-preserving?

sgn(αβ) =

{+1 if αβ is even αβ is even ⇔ α, β

−1 if αβ is odd both even or both odd

sgn(αβ) =

{+1 if α, β both even or both odd

−1 if one is even, the other odd

=

{+1 if sgn(α) = sgn(β) i.e. if sgn(α) = sgn(β) = ±1

−1 if sgn(α) 6= sgn(β) i.e. if sgn(α) = ±1, sgn(β) = ∓1

=

{+1 if sgn(α)sgn(β) = +1

−1 if sgn(α)sgn(β) = −1

= sgn(α)sgn(β) Thus sgn preserves the group operation

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 7 / 12

Page 13: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions to 3, continued

Recall: sgn(σ)def=

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Is sgn operation-preserving?

sgn(αβ) =

{+1 if αβ is even αβ is even ⇔ α, β

−1 if αβ is odd both even or both odd

sgn(αβ) =

{+1 if α, β both even or both odd

−1 if one is even, the other odd

=

{+1 if sgn(α) = sgn(β) i.e. if sgn(α) = sgn(β) = ±1

−1 if sgn(α) 6= sgn(β) i.e. if sgn(α) = ±1, sgn(β) = ∓1

=

{+1 if sgn(α)sgn(β) = +1

−1 if sgn(α)sgn(β) = −1

= sgn(α)sgn(β) Thus sgn preserves the group operation

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 7 / 12

Page 14: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions to 3, continued

Recall: sgn(σ)def=

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Is sgn operation-preserving?

sgn(αβ) =

{+1 if αβ is even αβ is even ⇔ α, β

−1 if αβ is odd both even or both odd

sgn(αβ) =

{+1 if α, β both even or both odd

−1 if one is even, the other odd

=

{+1 if sgn(α) = sgn(β) i.e. if sgn(α) = sgn(β) = ±1

−1 if sgn(α) 6= sgn(β) i.e. if sgn(α) = ±1, sgn(β) = ∓1

=

{+1 if sgn(α)sgn(β) = +1

−1 if sgn(α)sgn(β) = −1

= sgn(α)sgn(β) Thus sgn preserves the group operation

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 7 / 12

Page 15: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions to 3, continued

sgn(σ)def=

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Since sgn is a well-defined function that preserves the group operation, sgnis indeed a homomorphism.

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 8 / 12

Page 16: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions to 3, continuedKernel of sgn?

sgn(σ) =

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Recall: The kernel of a homomorphism is the set of all elements in thedomain that map to the identity of the range.

The identity of the multiplicative group {−1,+1} is 1.

Thus

Ker(sgn) = {α ∈ G |sgn(α) = 1}= {α ∈ G |α is even}

If G happens to be one of the Sn, then Ker(sgn) = An.

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 9 / 12

Page 17: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions to 3, continuedKernel of sgn?

sgn(σ) =

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Recall: The kernel of a homomorphism is the set of all elements in thedomain that map to the identity of the range.

The identity of the multiplicative group {−1,+1} is 1.

Thus

Ker(sgn) = {α ∈ G |sgn(α) = 1}= {α ∈ G |α is even}

If G happens to be one of the Sn, then Ker(sgn) = An.

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 9 / 12

Page 18: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Solutions to 3, continuedKernel of sgn?

sgn(σ) =

{+1 if σ is an even permutation,

−1 if σ is an odd permutation.

Recall: The kernel of a homomorphism is the set of all elements in thedomain that map to the identity of the range.

The identity of the multiplicative group {−1,+1} is 1.

Thus

Ker(sgn) = {α ∈ G |sgn(α) = 1}= {α ∈ G |α is even}

If G happens to be one of the Sn, then Ker(sgn) = An.Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 9 / 12

Page 19: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Properties of Homomorphisms

Let φ : G → G be a homomorphism, let g ∈ G , and let H ≤ G .

Properties of elements Properties of subgroups

1. φ(eG ) = eG 1. φ(H) ≤ G .2. φ(gn) = (φ(g))n for all n ∈ Z. 2. H cyclic =⇒ φ(H) cyclic.3. If |g | is finite, |φ(g)| divides |g |. 3. H Abelian =⇒ φ(H) Abelian.4. Ker(φ) ≤ G 4. H / G =⇒ φ(H) / φ(G )

7. K ≤ G =⇒ φ−1(K ) ≤ G .8. K / G =⇒ φ−1(K ) / G .

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 10 / 12

Page 20: Properties of Homomorphisms - Wheaton Collegejsklensky.webspace.wheatoncollege.edu/Abstract...Math 321-Abstract (Sklensky)In-Class WorkNovember 19, 2010 1 / 12 Let ˚: G !G be a homomorphism,

Properties of Homomorphisms

Let φ : G → G be a homomorphism, let g ∈ G , and let H ≤ G .

Properties of elements Properties of subgroups

1. φ(eG ) = eG 1. φ(H) ≤ G .2. φ(gn) = (φ(g))n for all n ∈ Z. 2. H cyclic =⇒ φ(H) cyclic.3. If |g | is finite, |φ(g)| divides |g |. 3. H Abelian =⇒ φ(H) Abelian.4. Ker(φ) ≤ G 4. H / G =⇒ φ(H) / φ(G )5. φ(a) = φ(b) ⇐⇒ aKer(φ) =bKer(φ)

5. |Ker(φ)| = n =⇒ φ is an n-to-1map

6. φ(g) = g ′ =⇒ φ−1(g ′) =gKer(φ)

6. |H| = n =⇒ |φ(H)| divides n

7. K ≤ G =⇒ φ−1(K ) ≤ G .8. K / G =⇒ φ−1(K ) / G .9. φ onto and Ker(φ) = {eG} =⇒φ an isomorphism.

Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 11 / 12

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Peer Review Exchange

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Christina, Z2[x ] ↔ Stephanie, GL(2,Z2)Laura, Z2[x ] ↔ Aubrie, C10

Tiffany, F → Eric, T2

Abbe, D6 → Rebecca, T2 → Roy, F↖←←← ← ←←←←←←←←← ← ←←←↩

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