Math 71
Class 11

Fermats last theorem
o One of the great mysteries of mathematics it has never been solved to date,
one of the worlds million dollar problems (if you find the solution or
proof you win a million dollars)
o X^n + y^n = z^n

Integral doma
HOMEWORK 2
MATH 71
DUE: 9/21/2012
1. (D&F 1.1.22, 25, 32) Orders. Let G be a group and let x G.
(a) If g G, show g 1 xg  = x and deduce that ab = ba for all a, b G.
Proof. Note that
(g 1 xg )n = g 1 x(gg 1 )x(gg 1 ) (gg 1 )xg = g 1 xn g
so that 1
Quiz 1, Math 71, Fall 2012
Justify all answers.
(1) Which of the following are groups. Why or why not?
(a) The positive rational numbers under multiplication.
(b) The negative rational numbers under multiplication.
(c) The invertible 2 2 matrices under th
HOMEWORK 3
MATH 71
DUE: 9/28/2012
SOLUTIONS
1.7.6 Prove that a group G acts faithfully on a set A if and only if the kernel of the action is trivial.
Proof. If the group acts faithfully, then the only element xing all of A is 1G , and so the
kernel is tri
HOMEWORK 4
MATH 71
DUE: 10/5/2012
SOLUTIONS
2.4.3 Prove that if H G is abelian, then H, Z (G) is abelian. Give an example where H, CG (H )
is not abelian.
Proof. If x H, Z (G) , then x = x11 x where xi H Z (G) and i = 1. Since the
elements of Z (G) commut
Quiz 2, Math 71, Fall 2012
Justify all answers. You may take for granted everything we have proved in class or on the homework
(including the fact that there are no normal subgroups of S4 of order 3 or 8), unless Im asking you
to briey justify a result yo
MIDTERM SOLUTIONS
(1) (20 pts) Let Zn = x  xn = 1 .
(a) Describe all homomorphisms : Zn Zn .
(b) Which of these homomorphisms are bijective?
(c) Show that the set
A = cfw_  : Zn Zn is a bijective homomorphism
is a group under function composition, and d
Quiz 3, Math 71, Fall 2012
Justify all answers. You may take for granted everything we have proved in class or on the homework
(including the fact that there are no normal subgroups of S4 of order 3 or 8), unless Im asking you
to briey justify a result yo
Name:
FINAL
MATH 71, FALL 2012
Due: November 19, 9am, 2012
Instructions:
Include this cover sheet, signed, and the problem sheet along with your solutions.
Complete this exam alone, without collaboration with others, and without the use of prohibited
reso
HOMEWORK 1
DUE: 9/14/2012
1. (D&F 0.1.5) Determine whether the following functions f are well dened:
(a) f : Q Z dened by f (a/b) = a.
(b) f : Q Q dened by f (a/b) = a2 /b2 .
Answer. Part (a) is not welldened because 1/2 = 2/4, but
f (1/2) = 1 = 2 = f (2