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)
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
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,
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
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 ) =