4
!;fr"'
6. (6 points) TRUE or FALSE. (If true, PROVE it. If false, give a COUNTER
EXAMPLE.) If G is a group and H = cfw_x3 I x E G , then H is a subgroup of
G.
F q(~cfw_
rdt
i)~ Lf
q ~Jtf~f
G?D:)
cfw_ T4~ H= [itJ/V; V"'fl~piJ
belQCe'JQ.; f (J h (f
~
5
and n be integers,
9. Let m
and let d be the greatest common
divisor of
"
ers rand s wlth d = rm + sn
n. Prove that there exists inte
g
~.
it
m and
.
LA
H =
cfw_
M t 4.
r
shif",f
fa
fcfw_S/1/"
t. It: fltR
A .J .~' &;'7 f,
Cl
H
"6
eLl'v;
~
1\ / r,
~
4
7. Let H be the subgroup
(124) of 84 , and let [x]
the element
~.'
cfw_(I), (12), (13), (23), (123), (132) of 84. Let x be
=
cfw_y E G I xyl
E H.
'1,
List the elements
of [x]. (Each element of [x] should appear in your list exactly once.)
1~c ek~(c

t/t;>@
2
4. TRUE or FALSE. (If true, PROVE it. If false, give a COUNTER
EXAMPLE.) Let G be a group and let a be a fixed element of G. If
'/
Pa: G t G , is the function which is given by Pa(g)
~
Y4~
then Pa is a permutation of the set G.
q /'5
fq
0.,
()~
6. TRUE or FALSE. (If true, PROVE it. If false, give a COUNTER
EXAMPLE.) If (G, *) is a group, then the function <p: 7 G , which
G
is given by <p(g) = 9 * 9 is a group homomorphism.
'./
fa lse
1~
(J :='~J,
We ~ee f~f
zCf3)
~(Jl, 1 ~.I,
cf(/< :;
2
"'
;'1
3. Is the group (Z;, x) a cyclic group? Why or why not?
1'7'r. hvJ
Je(c'"~Y~,
,4(so (2)2=.LlfJ.)
[;zJ3:.!)
();=C7)
f
~.
Q
I s~
cfw_Yt'~
5" e(t't:(1~fl.
QSS~l'J
f/7c
X
74rs
Llr
N
(i:)])
hcfw_(i 9i (~7
L ~144 eft '5 T~t!d("'1
tiu4< 2J);?I
cfw
Math 546,
Exam 2,
Solutions,
Fall 2011
Write everything on the blank paper provided.
You should KEEP this piece of paper.
If possible: turn the problems in order (use as much paper as necessary), use only
one side of each piece of paper, and leave 1 squar
MATH 546 Fall 2011 Notes about EXAM 1
1. Exam 1: Thursday, September 8.
The exam covers Homework sets
1,2,3,5,6,7,10,11. Be sure to MASTER all of the assigned homework.
2. The material on the old MATH 546 exams which is covered on your exam 1:
Exam 1s:
19
4
7. Let G be the group Ug. (a) LIST the elements of the set
'"
= cfw_g3 I9 E G.
S
'"
(b) Is S a subgroup of G? J
fa ~ G ]Jt h.5i~ ur
.*
"
iJ,
cr
~
. ustIfy your answer to (b).
l
1~ e(cfw_~"1/i
cT~ cit/!'l!;'1t"I(S"J
(I) >;: (
")
,
f=
(l./ f:=qJ
( u1l
2
3. Define "centralizer". Use complete sentences.
~
let
"'"
f
0
ca
be QI? e/e"'c~ f
;/J
G
~
f
bl
(Q
(q ~
1 f fR ufi C/4bt
I
~
T
'J
l.S d
Ii i~ c.let" il,;f C aJl lj q Sq~ Je:! ~ G.LIIetl~5T 5~v ev('I ~ 1'1<,cfw_.il<
c.n"~1'"
lc
I't~
Sa
."
r
;zbtq
]
;
~v~
Jcfw_X;l. E YQ''
~
~nry(!cfw_,
PRINT Your Name:
There are 10 problems on 5 pages. Each problem is worth 5 points.
'
/
I will put your exam outside my office door by noon on Friday. You may pick it up
any time before class on Monday. If I know your
~+(
[(Ct~ 1
JQ)(
.Yt/711(~
@
PRINT Your Name:
"'./'
There are 10 problems on ~pages. Each problem is worth 5 points.
1. Define "cyclic group". Use complete sentences.
The
9"O'tf
e.Ccfw_(l,.
G
/J q cYc.li( 'iJj,~p
(cfw_e;J~f
01 G
i.:(
(~ efC(e( 70
t~eW?
~
3
1"
'"
'"'.
~
6. Find all of the subgrou p s of TT  cfw_
.
b
t am that you have found allU9 the z Eel z9  I
.
cer
of
'> . b
su groups
'1 ~
U / ~!A
~ G, S q IL5'f,., J.7r.,
'
.
<
ex
I
V;
~
I
m
'f
I "'
4~ 4&)
lcfw_ 2 QI,i Cr7 Q"
e rele\3,
t ),c
@
4
8. TRUE or FALSE. (If true, PROVE it. If false, give a COUNTER EXAMPLE.)
If Hand K are subgroups of a group G, then the intersection H n K is also
a subgroup of G.
.~
Tvtfe

,
t'~ f if !c(ow /104
1*1[,
.rf Xj'iJ
I1Uo ~
f
?t"J Zr;f /( beCa~' t;'c,
Math 546,
Exam 2 SOLUTIONS,
Spring 2010
Write everything on the blank paper provided.
You should KEEP this piece of paper.
If possible: turn the problems in order (use as much paper as necessary), use only
one side of each piece of paper, and leave 1 squa
f1q1h S~b
Vt5
S! :),00(
PRINT Your Name:
There are 9 problems on 5 pages. Problem 1 is worth 12 points. Each of the other
problems is worth 11 points.
r.
1. Let 0" = (1,2,3)(4,5,6)
and T
= (3,4,5)
be elements of 86 . Write
~
TO"Tl as
the pro5~uct,of disj
4
5. Let U8 = cfw_z E <C z8 = I. Which elements in U8 have the form x3 for some
x E U8? Explain your answer.
I
.
~
(f:~td
W
t.3
dj Vi
elt~
tcfw_ ;. 0S :lIT 1 i 51?
=(
3cfw_
h"j ~
:'
,'6"
(u13 : U3
~th
f~
X3 tot
~t)~
r
X
lY;)
q3.
Tcfw_
.
l tI f?> : Uf,
2
3. TRUE or FALSE. (If true, PROVE it. If false, give a COUNTER EXAMPLE.)
If every proper subgroup H of a group G is abelian, then G is abelian.
6rB
'/
.
TAt drDl1f S3
FCcfw_ /se
~
AClJ
1
It(JICcfw_?J,(~
6
53
l"Oot('~
~cfw_('e
fe (J
tt
~
OJ~
'f'lt1
f
I
3
5. TRUE or FALSE. (If true, PROVE it. If false, give a COUNTER EXAMPLE.)
If A is a finite set and b is an element of A, then
'"
T
;:. cfw_o E SA I o(b)
= b
IS a group.
Trut?c:.:(d,IIvl'
,
r,~(~J
.::
r~
V, 'l' 
it
I
trot (k)
(Ie T i ~
a(Io) ~
h
2
3. Let G be a group and a E G .
(a) Define "the centralizer of a".
T~(! (J:.'~1/f6it:t. af f~(
~c ( <!)f
,'
a.lR.
e(t!'/C~ f q
e(!*,(1~ iJ
1<. tcJ't/ Gj
l&t
IS tv
(,v~(~cA C<!)/(J/7~
fe
cfw_'I G
l,vt'I~ Q.
,
.I
(b) Prove that the centralizer of
3
4.
'"
(a) Give an example of a cyclic subgroup of D4 of order 4.
necessary.
<f f 'J
= cfw_41/
t f~ f
No proof is
!>]
/
~
(b) Give an example of a subgroup of D4 of order 4 which is not cyclic. No
proof is necessary.
i0J
fl/ v;
Vf2]
Math 546, Exam 1, Spring 2010
Write everything on the blank paper provided. You should KEEP this piece of
paper. If possible: turn the problems in order (use as much paper as necessary),
use only one side of each piece of paper, and leave 1 square inch in
Math 546
Homework 10
1
546 Problems.
11.2 Show that the subgroup
a b
H=
: ad 6= 0 GL(2, R)
0 d
is not normal.
0 1
0
1
Proof: We observe that S =
GL(2, R), that S
=
1
0
1
1 1
T =
H. We compute
0 1
0 1
1 1
0 1
0 1
0 1
1
1
ST S =
=
=
1 0
0 1
1 0
1 1
1
Math 546
Homework 3
1
546 Problems.
4.16. Suppose that G = hxi is a cyclic group. Prove that G = hx1 i.
Proof: We observe that G = hxi = cfw_. . . , x2 , x1 , e, x, x2 , . . . , and that
hx1 i = cfw_. . . , (x1 )2 , (x1 )1 , e, x1 , x2 , . . . = cfw_. .
Math 546
Homework 4
1
546 Problems.
5.1 g) Determine whether or not H = cfw_I, J, K is a subgroup of Q8 .
Claim: H is not a subgroup of Q8 .
Proof: H is not a subgroup since it is not closed. For example, J 2 = I 6 H.
5.4 c)
(a) How many subgroups does (Z
Math 546
Homework 1
1.3 and 1.6. For each of the following sets S and functions on SS, determine whether is
a binary operation on S. If is a binary operation on S, determine whether it is commutative
and whether it is associative.
a) S = Z, a b = a + b2 .