Download Math 3121 Lecture 12

Math 3121
Abstract Algebra I
Lecture 12
Finish Section 14
Review
Next Midterm
• Midterm 2 is Nov 13.
Covers sections: 7-14 (not 12)
Review on Thursday
Cosets of a Homomorphism
Theorem: Let h: G  G’ be a group
homomorphism with kernel K. Then the
cosets of K form a group with binary operation
given by (a K)(b K) = (a b) K. This group is
called the factor group G/K. Additionally, the
map μ that takes any element x of G to is
coset xH is a homomorphism. This is called
the canonical homomorphism.
Coset Multiplication is equivalent to Normality
Theorem: Let H be a subgroup of a group
G. Then H is normal if and only if
(a H )( b H) = (a b) H, for all a, b in G
Canonical Homomorphism Theorem
Theorem: Let H be a normal subgroup of a
group G. Then the canonical map : G  G/H
given by (x) = x H is a homomorphism with
kernel H.
Proof: If H is normal, then by the previous
theorem, multiplication of cosets is defined
and  is a homomorphism.
Fundamental Homomorphism Theorem
Theorem: Let h: G  G’ be a group homomorrphism
with kernel K. Then h[G] is a group, and the map μ:
G/K  h[G] given by μ(a K) = h(a) is an isomorphism.
Let : G  G/H be the canonical map given by (x) = x
H. Then h = μ .
h
G
h[G]
μ

G/Ker(h)
Proof of Fundamental Thoerem
• Proof: This theorem just gathers together what we
have already shown. We have already shown that
h[G] is a group. We have h(a) = h(b) iff aK = bK. Thus
μ exists. μ((x)) = μ(x H) = h(x).
x
h(x)
h
G
h[G]
μ

x Ker(h)
G/Ker(h)
Properties of Normal Subgroups
Theorem: Let H be a subgroup of a group G. The following conditions are equivalent:
1) g h g-1  H, for all g in G and h in H
2) g H g-1 = H, for all g in G
3) g H = H g, for all g in G
Proof:
1) ⇒ 2): H  g H g-1
1) ⇒ g H g-1  H ⇒ g H g-1  H and
g H g-1  H ⇒ 2)
2) ⇒ 3):
Assume 2). Then x in g H ⇒ x g-1 in H ⇒ x in H g
and x in H g ⇒ x g-1 in H ⇒ g x g-1 in g H
3) ⇒ 1):
Assume 3). Then h  H ⇒ g h  g H ⇒ g h  H g ⇒ g h g-1  H
Automorphism
Definition: An isomorphism of a group with
itself is called an automorhism
Definition: The automorphism ig: G  G given
by ig (x) = g x g-1 is the inner automorphism of
G by g. This sometimes called conjugation of x
by g.
Note: ig is an automorphism.
More Terminology
• Invariant subgroups
• Congugate subgroup. – examples in S3
HW: Section 14
• Don’t hand in
Pages 142-143: 1, 3, 5, 9, 11, 25, 29, 31
• Hand in:
Pages 142-143: 24, 37
Review