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 Hand in: Pages 142-143: 1, 3, 5, 9, 11, 25, 29, 31 Hand in: Pages 142-143: 24, 37

Review