1. Math 3121 Abstract Algebra I
Lecture 12
Finish Section 14
Review

2. Next Midterm Midterm 2 is Nov 13. Covers sections: 7-14 (not 12)
Review on Thursday

3. 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.

4. 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

5. 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.

6. 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)

7. 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)

8. 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

9. 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.

10. More Terminology Invariant subgroups
Congugate subgroup. – examples in S3

11. HW: Section 14 Don’t hand in Hand in:
Pages : 1, 3, 5, 9, 11, 25, 29, 31
Hand in:
Pages : 24, 37

12. Review