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.
Finish Section 14
Covers sections: 7-14 (not 12)
Review on Thursday
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.
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
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.
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.
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
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
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.
Pages 142-143: 1, 3, 5, 9, 11, 25, 29, 31
Pages 142-143: 24, 37