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) / φ(G )
8.
¯
K /
¯
G =⇒ φ
−1
(
¯
K ) / G .
Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 1 / 12
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(e
G
) = e
¯
G
1. φ(H) ≤
¯
G .
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.
¯
K ≤
¯
G =⇒ φ
−1
(
¯
K ) ≤ G .
8.
¯
K /
¯
G =⇒ φ
−1
(
¯
K ) / G .
Remember that
Ker(φ)
def
= {g ∈ G |φ(g ) = id
¯
G
}.
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) = (e
G
, 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 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) = e
G
}
= {(g, h)
g = e
G
}
= {(e
G
, h)
h ∈ H}
= {e
G
} ⊕ 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) = (e
G
, h).
Ker(i)
def
= {h ∈ H
i(h) = e
G ⊕H
= (e
G
, e
H
)}
= {h ∈ H
(e
G
, h) = (e
G
, e
H
)}
= {e
H
}.
Or...
Since i is a homomorphism, i(e
H
) = e
G ⊕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
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 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
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 multiplicative
group {+1, −1}. What is the kernel?
Is sgn well-defined?
Suppose that σ
1
= σ
2
.
Then since every permutation’s factorization into transpositions will be
either always odd or always even, either both σ
1
and σ
2
are even or both
σ
1
and σ
2
are odd.
Thus sgn(σ
1
) = sgn(σ
2
), and so sgn is a well-defined function.
Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 6 / 12
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
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
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
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
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
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
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
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, sgn
is indeed a homomorphism.
Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 8 / 12
Solutions to 3, continued
Kernel 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 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 S
n
, then Ker(sgn) = A
n
.
Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 9 / 12
Solutions to 3, continued
Kernel 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 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 S
n
, then Ker(sgn) = A
n
.
Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 9 / 12
Solutions to 3, continued
Kernel 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 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 S
n
, then Ker(sgn) = A
n
.
Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 9 / 12
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(e
G
) = e
¯
G
1. φ(H) ≤
¯
G .
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.
¯
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
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(e
G
) = e
¯
G
1. φ(H) ≤
¯
G .
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 )
5. φ(a) = φ(b) ⇐⇒ aKer(φ) =
bKer(φ)
5. |Ker(φ)| = n =⇒ φ is an n-to-1
map
6. φ(g) = g
0
=⇒ φ
−1
(g
0
) =
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) = {e
G
} =⇒
φ an isomorphism.
Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 11 / 12
Peer Review Exchange
Luke, gps of order 8 ↔ Marie, gps of order 9
Dan, Q
4
↔ Sam, Z[i]
Alfred, M ↔ Becky H
Shawn, U(13), U(21) ↔ Erin, Q
2,3
Christina, Z
2
[x] ↔ Stephanie, GL(2, Z
2
)
Laura, Z
2
[x] ↔ Aubrie, C
10
Tiffany, F → Eric, T
2
Abbe, D
6
→ Rebecca, T
2
→ Roy, F
-←←← ← ←←←←←←←←← ← ←←←-
Math 321-Abstract (Sklensky) In-Class Work November 19, 2010 12 / 12