Download Lecture 10 More on quotient groups

Lecture 10
More on quotient groups
Last time
The fibers of a homomorphism of groups can form their own group. In
fact, the precise range of the homomorphism can be forgotten. We’re
figuring out how to use the coset structure of those fibers to motivate
forgetting the homomorphism as well (i.e. we’re looking for an internal
criterion on subgroups of G for defining quotients of G).
Last time
The fibers of a homomorphism of groups can form their own group. In
fact, the precise range of the homomorphism can be forgotten. We’re
figuring out how to use the coset structure of those fibers to motivate
forgetting the homomorphism as well (i.e. we’re looking for an internal
criterion on subgroups of G for defining quotients of G).
Let H ≤ G, and fix g ∈ G. The set
gH = {gh | h ∈ H}
is a left coset of H. Any element of gH is called a representative.
Last time
The fibers of a homomorphism of groups can form their own group. In
fact, the precise range of the homomorphism can be forgotten. We’re
figuring out how to use the coset structure of those fibers to motivate
forgetting the homomorphism as well (i.e. we’re looking for an internal
criterion on subgroups of G for defining quotients of G).
Let H ≤ G, and fix g ∈ G. The set
gH = {gh | h ∈ H}
is a left coset of H. Any element of gH is called a representative.
Some facts about cosets:
1. The cosets of H partition G into equal sized sets.
2. As a consequence, if y ∈ xH, then xH = yH.
3. Two cosets xH and yH are equal exactly when x−1 y ∈ H.
4. The fibers of any homomorphism are both left and right
cosets of the kernel.
The question we ended on last time:
When do the set of cosets of a set form a group?
i.e. when is the multiplication
xH ? yH = (xy)H
well-defined?
The question we ended on last time:
When do the set of cosets of a set form a group?
i.e. when is the multiplication
xH ? yH = (xy)H
well-defined?
Proposition
Let H ≤ G.
1. The operation ? above is well defined if and only if
gxg −1 ∈ H for all x ∈ H and g ∈ G. (In other words, when
the normalizer NG (H) = G)
2. If ? is well-defined, then G/H = {gH | g ∈ G} forms a group
with 1 = 1H and (gH)−1 = g −1 H.
The question we ended on last time:
When do the set of cosets of a set form a group?
i.e. when is the multiplication
xH ? yH = (xy)H
well-defined?
Proposition
Let H ≤ G.
1. The operation ? above is well defined if and only if
gxg −1 ∈ H for all x ∈ H and g ∈ G. (In other words, when
the normalizer NG (H) = G)
2. If ? is well-defined, then G/H = {gH | g ∈ G} forms a group
with 1 = 1H and (gH)−1 = g −1 H.
Definition
A subgroup N ≤ G is normal if gN g −1 = N for all g ∈ G, i.e.
when G/N is a well-defined group. Write N E G.
In summary, let N ≤ G. Then the following are equivalent:
1. N is normal in G
2. NG (N ) = G
3. gN = N g for all g ∈ G
4. the operation on left cosets given by xN ? yN = (xy)N is
well-defined
5. gN g −1 ⊆ N for all g ∈ G
Example:
Letting D8 act on itself by conjugation (g · a = gag −1 ) yields the
following table (from Lecture 6):
act on ↓
act by →
1
r
r2
r3
s
sr
sr2
sr3
1
1
1
1
1
1
1
1
1
r
r
r
r
r
r3
r3
r3
r3
r2
r2
r2
r2
r2
r2
r2
r2
r2
r3
r3
r3
r3
r3
r
r
r
r
What are the normal subgroups of D8 ?
s
s
sr2
s
sr2
s
sr2
s
sr2
sr
sr
sr3
sr
sr3
sr3
sr
sr3
sr
sr2
sr2
s
sr2
s
sr2
s
sr2
s
sr3
sr3
sr
sr3
sr
sr
sr3
sr
sr3
Example:
Letting D8 act on itself by conjugation (g · a = gag −1 ) yields the
following table (from Lecture 6):
act on ↓
act by →
1
r
r2
r3
s
sr
sr2
sr3
1
1
1
1
1
1
1
1
1
r
r
r
r
r
r3
r3
r3
r3
r2
r2
r2
r2
r2
r2
r2
r2
r2
r3
r3
r3
r3
r3
r
r
r
r
s
s
sr2
s
sr2
s
sr2
s
sr2
sr
sr
sr3
sr
sr3
sr3
sr
sr3
sr
sr2
sr2
s
sr2
s
sr2
s
sr2
s
sr3
sr3
sr
sr3
sr
sr
sr3
sr
sr3
What are the normal subgroups of D8 ?
Note: “subgroup” is transitive but “normal” is not!! For example,
{1, s} E {1, r2 , s, sr2 } and {1, r2 , s, sr2 } E D8 , but {1, s} 5 D8 .
More examples
1. G and 1 = {1G } are always normal in G, with
G/1 ∼
=G
and
G/G ∼
= 1.
More examples
1. G and 1 = {1G } are always normal in G, with
G/1 ∼
=G
and
G/G ∼
= 1.
2. All subgroups of the center Z(G) of a group are normal in G.
More examples
1. G and 1 = {1G } are always normal in G, with
G/1 ∼
=G
and
G/G ∼
= 1.
2. All subgroups of the center Z(G) of a group are normal in G.
3. All subgroups of abelian groups are normal.
More examples
1. G and 1 = {1G } are always normal in G, with
G/1 ∼
=G
and
G/G ∼
= 1.
2. All subgroups of the center Z(G) of a group are normal in G.
3. All subgroups of abelian groups are normal.
4. Quotients of abelian groups are abelian and quotients of cyclic
groups are cyclic.
Back to homomorphism
Let N E G. The map
π : G → G/N
g 7→ gN
is a homomorphism, and is called the natural projection of G onto
G/N .
Back to homomorphism
Let N E G. The map
π : G → G/N
g 7→ gN
is a homomorphism, and is called the natural projection of G onto
G/N .
Proposition
A subgroup N of G is normal if and only if it is the kernel of some
homomorphism.
Back to homomorphism
Let N E G. The map
π : G → G/N
g 7→ gN
is a homomorphism, and is called the natural projection of G onto
G/N .
Proposition
A subgroup N of G is normal if and only if it is the kernel of some
homomorphism.
Theorem (First isomorphism theorem)
If ϕ : G → H is a homomorphism of groups, then
G/ker(ϕ) ∼
= ϕ(G).
Back to homomorphism
Let N E G. The map
π : G → G/N
g 7→ gN
is a homomorphism, and is called the natural projection of G onto
G/N .
Proposition
A subgroup N of G is normal if and only if it is the kernel of some
homomorphism.
Theorem (First isomorphism theorem)
If ϕ : G → H is a homomorphism of groups, then
G/ker(ϕ) ∼
= ϕ(G).
Punchline: The study of the isomorphism classes of homomorphic
images of G is “the same” as the study of the normal subgroups of G.