1) Let f1 , f2 S4 be dened as:
f2 : 1 3
21
32
44
f1 : 1 2
21
34
43
(a) Find f2 f1 and f2 1 . [Your answer should be given in the same form as f1 and f2 are
given above.]
Solution.
f2 1 : 1 2
23
31
44
f2 f1 : 1 f2 (f1 (1) = f2 (2) = 1
2 f2 (f1 (2) = f2 (1)
1) Suppose that |G| = 2p, where p is a prime dierent from 2. Prove that either G C2p or
=
G D2p .
=
Proof. By the First Sylow Theorem, [since 2 and p are both primes and p = 2] there are
subgroups H and K such that |H | = p and |K | = 2. Hence, since they
1) Give the conjugacy classes and the class equation for Q8 . [Hint: Let Q8 act on itself by
conjugation. Then the conjugacy classes are the distinct orbits, and the class equation is given
by the orders of these classes. The class equation is something l
Extra Credit Problem
(Due in class on Monday 10/30.)
Math 455
Problem from this years UT Math Contest (Fermat II) for high school students.
Problem: Let a, b, c cfw_1, 2, . . . , 2005 and
def
f (X ) = aX 101 + bX 100 + c.
Prove that if f (2006) is prime,
def
def
1) Let G = Z/36Z and H = . [As usual, a represents the coset (a + 36Z) of Z/36Z.]
2
3
(a) Describe G/H as a set. [In other words, give its elements.]
Solution. We have that:
= cfw_, , , . . . , 32
2
024
3 = cfw_, , , . . . , 33
036
Thus,
= = cfw
def
1) Let G = C4 C8 . [As usual, Cn denotes the cyclic group of order n.] Let x and y denote
def
the generators of C4 and C8 respectively, i.e., C4 = x and C8 = y , and let H = (x, y 7 ) .
(a) Give the elements of H explicitly.
Solution.
H = (x, y 7 ) =