1. Math 3121 Abstract Algebra I
Lecture 13
Midterm 2 back and over
And Start Section 15

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

3. Midterm 2
Midterm 2 is back and over

4. Section 15: Factor Groups
Examples of factor groups
When G/H has order 2 – H must be normal
Falsity of converse to Lagrange’s Theorem – example A4
ℤn×ℤm/<(0,1)>
G1×G2/i1 (G1) and G1×G2/i2 (G2)
ℤ4×ℤ6/<(2,3)>
Th: Factor group of a cyclic group is cyclic
Th: Factor group of a finitely generated group is finitely generated.
Def: Simple groups
Alternating group An, for 5≤ n, is simple (exercise 39)
Preservation of normality via homomorphisms
Def: Maximal normal subgroup
Th: M is a maximal normal subgroup of G iff G/M is simple
Def: Center
Def: Commutator subgroup

5. Examples of Factor Groups
G/{e} isomorphic to G
G/G isomorphic to {e}

6. Index 2 Subgroups are normal
Theorem: If H is a subgroup of index 2 in a group finite group G, then H is normal. Proof: H and G-H partition the group G in halves. Thus G-H must be the left and the right coset of H in G other than H itself. Thus all left cosets are right cosets.

7. Falsity of the Converse of Lagrange’s Theorem
Example: A4 Theorem: A4 has order 12, but none of its subgroups has order 6. Proof: The elements of G=A4 are (), (1 2 3), (1 3 2), (1 2 4), (1 4 2), (1 3 4), (1 4 3), (2 3 4), (2 4 3), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3). Suppose H is a subgroup of order 6 of A4. Then G/H has two elements H and a H, for some a not in H. It is isomorphic to Z2. Its multiplication table has H H = H and (a H) (a H) = H. Thus x2 is in H for all x in G. Note that (1 2 3)2 = (1 3 2), (1 3 2)2 = (1 2 3), etc. Thus all three cycles are squares and hence are in H. However, there are 8 of them. So order H is larger that 6.