Transcript Math 3121 Lecture 13

Math 3121
Abstract Algebra I
Lecture 13
Midterm 2 back and over
And Start Section 15
HW: Section 14
• Don’t hand in
Pages 142-143: 1, 3, 5, 9, 11, 25, 29, 31
• Hand in (Tues Nov 25):
Pages 142-143: 24, 37
Midterm 2
• Midterm 2 is back and over
Section 15: Factor Groups
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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
Examples of Factor Groups
• G/{e} isomorphic to G
• G/G isomorphic to {e}
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.
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.