Properties of Homomorphisms
Recall: A function φ : G → Ḡ is a homomorphism if
φ(ab) = φ(a)φ(b)∀a, b ∈ G .
Let φ : G → Ḡ be a homomorphism, let g ∈ G , and let H ≤ G .
Properties of elements
Properties of subgroups
1. φ(eG ) = eḠ
1. φ(H) ≤ Ḡ .
2. φ(g n ) = (φ(g ))n ∀n ∈ Z.
3. If |g | is finite, |φ(g )| |g |.
2. H cyclic =⇒ φ(H) cyclic.
3. H Abelian =⇒ φ(H) Abelian.
7. K̄ ≤ Ḡ =⇒ φ−1 (K̄ ) ≤ G .
4. H / G =⇒ φ(H) / φ(G )
8. K̄ / Ḡ =⇒ φ−1 (K̄ ) / G .
Math 321-Abstract (Sklensky)
In-Class Work
November 19, 2010
1 / 12
Properties of Homomorphisms
Let φ : G → Ḡ be a homomorphism, let g ∈ G , and let H ≤ G .
Properties of elements
Properties of subgroups
1. φ(eG ) = eḠ
1. φ(H) ≤ Ḡ .
2. φ(g n ) = (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. K̄ ≤ Ḡ =⇒ φ−1 (K̄ ) ≤ G .
8. K̄ / Ḡ =⇒ φ−1 (K̄ ) / G .
Remember that
def
Ker (φ) = {g ∈ G |φ(g ) = idḠ }.
Math 321-Abstract (Sklensky)
In-Class Work
November 19, 2010
2 / 12
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 by
i(h) = (eG , h).
3. Let G be a group of permutations. For each σ ∈ G , define
(
+1 if σ is an even permutation,
sgn(σ) =
−1 if σ is an odd permutation.
Prove that sgn is a homomorphism from G to the multiplicative
group {+1, −1}. What is the kernel?
Math 321-Abstract (Sklensky)
In-Class Work
November 19, 2010
3 / 12
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
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 can
map to the identity. Thus the kernel, which is the set of all things that
map to the identity, contains only the identity.
Notice: This shows that the kernel of any 1-1 homomorphism consists only
of the identity!
Math 321-Abstract (Sklensky)
In-Class Work
November 19, 2010
5 / 12
Solutions:
3. Let G be a group of permutations. For each σ ∈ G , define
(
+1 if σ is an even permutation,
sgn(σ) =
−1 if σ is an odd permutation.
Prove that sgn is a homomorphism from G to the multiplicative
group {+1, −1}. What is the kernel?
To show that sgn is a homomorphism, NTS sgn is a well-defined function
and is operation-preserving.
Math 321-Abstract (Sklensky)
In-Class Work
November 19, 2010
6 / 12
Solutions:
3. Let G be a group of permutations. For each σ ∈ G , define
(
+1 if σ is an even permutation,
sgn(σ) =
−1 if σ is an odd permutation.
Prove that sgn is a homomorphism from G to the multiplicative
group {+1, −1}. What is the kernel?
Is sgn well-defined?
Suppose that σ1 = σ2 .
Then since every permutation’s factorization into transpositions will be
either always odd or always even, either both σ1 and σ2 are even or both
σ1 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
Solutions to 3,
( continued
def
Recall: sgn(σ) =
+1
−1
if σ is an even permutation,
if σ is an odd permutation.
Is sgn operation-preserving?
(
+1 if αβ is even
sgn(αβ) =
−1 if αβ is odd
Math 321-Abstract (Sklensky)
In-Class Work
November 19, 2010
7 / 12
Solutions to 3,
( continued
def
Recall: sgn(σ) =
+1
−1
if σ is an even permutation,
if σ is an odd permutation.
Is sgn operation-preserving?
(
+1 if αβ is even
sgn(αβ) =
−1 if αβ is odd
Math 321-Abstract (Sklensky)
αβ is even ⇔ α, β
both even or both odd
In-Class Work
November 19, 2010
7 / 12
Solutions to 3,
( continued
def
Recall: sgn(σ) =
+1
−1
if σ is an even permutation,
if σ is an odd permutation.
Is sgn operation-preserving?
(
αβ is even ⇔ α, β
+1 if αβ is even
sgn(αβ) =
−1 if αβ is odd
both even or both odd
(
+1 if α, β both even or both odd
sgn(αβ) =
−1 if one is even, the other odd
Math 321-Abstract (Sklensky)
In-Class Work
November 19, 2010
7 / 12
Solutions to 3,
( continued
def
Recall: sgn(σ) =
+1
−1
if σ is an even permutation,
if σ is an odd permutation.
Is sgn operation-preserving?
(
+1
sgn(αβ) =
−1
(
+1
sgn(αβ) =
−1
(
+1
=
−1
Math 321-Abstract (Sklensky)
if αβ is even
if αβ is odd
αβ is even ⇔ α, β
both even or both odd
if α, β both even or both odd
if one is even, the other odd
if sgn(α) = sgn(β)
if sgn(α) =
6 sgn(β)
In-Class Work
November 19, 2010
7 / 12
Solutions to 3,
( continued
def
Recall: sgn(σ) =
+1
−1
if σ is an even permutation,
if σ is an odd permutation.
Is sgn operation-preserving?
(
+1
sgn(αβ) =
−1
(
+1
sgn(αβ) =
−1
(
+1
=
−1
Math 321-Abstract (Sklensky)
if αβ is even
if αβ is odd
αβ is even ⇔ α, β
both even or both odd
if α, β both even or both odd
if one is even, the other odd
if sgn(α) = sgn(β)
if sgn(α) =
6 sgn(β)
In-Class Work
i.e. if sgn(α) = sgn(β) = ±1
i.e. if sgn(α) = ±1, sgn(β) = ∓1
November 19, 2010
7 / 12
Solutions to 3,
( continued
def
Recall: sgn(σ) =
+1
−1
if σ is an even permutation,
if σ is an odd permutation.
Is sgn operation-preserving?
sgn(αβ) =
sgn(αβ) =
=
=
(
+1
−1
(
+1
−1
(
+1
−1
(
+1
−1
Math 321-Abstract (Sklensky)
if αβ is even
if αβ is odd
αβ is even ⇔ α, β
both even or both odd
if α, β both even or both odd
if one is even, the other odd
if sgn(α) = sgn(β)
if sgn(α) =
6 sgn(β)
i.e. if sgn(α) = sgn(β) = ±1
i.e. if sgn(α) = ±1, sgn(β) = ∓1
if sgn(α)sgn(β) = +1
if sgn(α)sgn(β) = −1
In-Class Work
November 19, 2010
7 / 12
Solutions to 3,
( continued
def
Recall: sgn(σ) =
+1
−1
if σ is an even permutation,
if σ is an odd permutation.
Is sgn operation-preserving?
sgn(αβ) =
sgn(αβ) =
=
=
(
+1
−1
(
+1
−1
(
+1
−1
(
+1
−1
if αβ is even
if αβ is odd
if α, β both even or both odd
if one is even, the other odd
if sgn(α) = sgn(β)
if sgn(α) =
6 sgn(β)
i.e. if sgn(α) = sgn(β) = ±1
i.e. if sgn(α) = ±1, sgn(β) = ∓1
if sgn(α)sgn(β) = +1
if sgn(α)sgn(β) = −1
= sgn(α)sgn(β)
Math 321-Abstract (Sklensky)
αβ is even ⇔ α, β
both even or both odd
Thus sgn preserves the group operation
In-Class Work
November 19, 2010
7 / 12
Solutions to 3, continued
(
+1
sgn(σ) =
−1
def
if σ is an even permutation,
if σ is an odd permutation.
Since sgn is a well-defined function that preserves the group operation, sgn
is indeed a homomorphism.
Math 321-Abstract (Sklensky)
In-Class Work
November 19, 2010
8 / 12
Solutions to 3, continued
Kernel of sgn?
(
+1
sgn(σ) =
−1
if σ is an even permutation,
if σ is an odd permutation.
Recall: The kernel of a homomorphism is the set of all elements in the
domain that map to the identity of the range.
The identity of the multiplicative group {−1, +1} is 1.
Math 321-Abstract (Sklensky)
In-Class Work
November 19, 2010
9 / 12
Solutions to 3, continued
Kernel of sgn?
(
+1
sgn(σ) =
−1
if σ is an even permutation,
if σ is an odd permutation.
Recall: The kernel of a homomorphism is the set of all elements in the
domain 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}
Math 321-Abstract (Sklensky)
In-Class Work
November 19, 2010
9 / 12
Solutions to 3, continued
Kernel of sgn?
(
+1
sgn(σ) =
−1
if σ is an even permutation,
if σ is an odd permutation.
Recall: The kernel of a homomorphism is the set of all elements in the
domain 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
Properties of Homomorphisms
Let φ : G → Ḡ be a homomorphism, let g ∈ G , and let H ≤ G .
Properties of elements
1. φ(eG ) = eḠ
2. φ(g n ) = (φ(g ))n for all n ∈ Z.
3. If |g | is finite, |φ(g )| divides |g |.
4. Ker (φ) ≤ G
Math 321-Abstract (Sklensky)
Properties of subgroups
1. φ(H) ≤ Ḡ .
2. H cyclic =⇒ φ(H) cyclic.
3. H Abelian =⇒ φ(H) Abelian.
4. H / G =⇒ φ(H) / φ(G )
7. K̄ ≤ Ḡ =⇒ φ−1 (K̄ ) ≤ G .
8. K̄ / Ḡ =⇒ φ−1 (K̄ ) / G .
In-Class Work
November 19, 2010
10 / 12
Properties of Homomorphisms
Let φ : G → Ḡ be a homomorphism, let g ∈ G , and let H ≤ G .
Properties of elements
1. φ(eG ) = eḠ
2. φ(g n ) = (φ(g ))n for all n ∈ Z.
3. If |g | is finite, |φ(g )| divides |g |.
4. Ker (φ) ≤ G
5. φ(a) = φ(b) ⇐⇒ aKer (φ) =
bKer (φ)
6. φ(g ) = g 0 =⇒ φ−1 (g 0 ) =
gKer (φ)
Properties of subgroups
1. φ(H) ≤ Ḡ .
2. H cyclic =⇒ φ(H) cyclic.
3. H Abelian =⇒ φ(H) Abelian.
4. H / G =⇒ φ(H) / φ(G )
5. |Ker (φ)| = n =⇒ φ is an n-to-1
map
6. |H| = n =⇒ |φ(H)| divides n
7. K̄
8. K̄
9. φ
φ an
Math 321-Abstract (Sklensky)
In-Class Work
≤ Ḡ =⇒ φ−1 (K̄ ) ≤ G .
/ Ḡ =⇒ φ−1 (K̄ ) / G .
onto and Ker (φ) = {eG } =⇒
isomorphism.
November 19, 2010
11 / 12
Peer Review Exchange
Abbe, D6
-←←←
→
←
Luke, gps of order 8
Dan, Q4
Alfred, M
Shawn, U(13), U(21)
Christina, Z2 [x]
Laura, Z2 [x]
Tiffany, F
Rebecca, T2
←←←←←←←←←
Math 321-Abstract (Sklensky)
In-Class Work
↔
↔
↔
↔
↔
↔
→
→
←
Marie, gps of order 9
Sam, Z[i]
Becky H
Erin, Q2,3
Stephanie, GL(2, Z2 )
Aubrie, C10
Eric, T2
Roy, F
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