MAS 41301-Practir:er problern Set #L
1. (o) Give an exanrltJie ol a.rr al_,elian group G whic:h is not cyciic.
(b,) l,et G be iL gl"our:) tr,nd Lert r.t; , (1. prol,e that (ry)-r t1-Lr*t.
(c,) f,et G lle a g]'or:.t1-, Provt. that (l is :rlrcli:in if :r'cl
/, (nl G;re a/.e?(.a/a/,fu
f^tl ,/h/art grtu/ 6 rA./bd?ilL
,Gf V= 4l e,q,cfw_t qb 4,118 a /'re.
" non-,cfw_h/an f^/.
-Z/ k d
sincz )f druo .rtt rath)n
/tt e k ^,fr$, /*
A"/ pcg )-,
Summer C 2016
The following problems are due on Friday, May 20, at the beginning of class.
a. Use Euclids Algorithm to find the gcd(36, 210).
b. Find integers x, y such that gcd(36, 210) = 36x + 210y.
2. Recall that Zn = c
Final Exam Review
Summer C 2016
The final exam will be in class on Friday, August 5. The exam will cover Chapters 9-14 of
the textbook (with the exception of characteristic of a ring in Ch.13). I follow the numbering
from the 8th edition of the
Basic proof patterns
Direct proof. Always list the assumption at the beginning of the proof, and list the conclusion at the
end of the proof. Fill in the logical steps in between.
Proof by contradiction. Always list the assumpti
Exam 2 Material
Key Notes from Chapter 5
S n has order n!
S n is non-Abelian when n 3
disjoint cycle: the various cycles have no number in common
Let = 1 2 , r be a cycle decomposition of . Thus
i is a k icycle ,supp ( i ) su
Definition Coset of H in G
Let G be a grouplet H be a nonemptyof G . For any g G , the set cfw_ gh|h H is denoted by gH .
When H is a subgroup of G , the set gH is called the of H G containing g .
g is called the coset representative of gH.
ABSTRACT ALGEBRA STUDY QUESTIONS III
1. Dene the center Z (G) and show that Z (G) is a normal subgroup of G.
2. Show that if |G|/|H | = 2, then H is normal in G.
3. Let H be a normal subgroup of G. Show that G/H is a group.
4. If H is the only subgroup of
ABSTRACT ALGEBRA STUDY QUESTIONS I
1. List the 8 elements of the group D4 as reections and rotations. Indicate the
identity element and the inverse of each element. Give an example to show that
D4 is not commutative. Also represent D4 as a subgroup of S4
ABSTRACT ALGEBRA STUDY QUESTIONS II
and nd the order of .
1. Express =
as a product of disjoint cycles
2. Express = (1, 2, 4, 8)(3, 6)(7, 10, 13) as a product of transpositions and
determine whether is odd or even.
Keys Notes from Abstract Chapter 3
G L2 (R) = cfw_A 2x2 matrices| det(A) 0
S L2 (R) = cfw_A 2x2 matrices| det(A) 1
The center of a group Z(G) is always non empty because e will always
be an element of the center.
The centralizer of e in G, C(e), is e