MATH 241 Exam 2, Fall 2010
American University of Beirut · Introduction to Abstract Algebra · 8 pages
More MATH 241 files
Show the text of this exam
Math 241
Name:
Student ID:
Instructions:
1. This exam has 6 pages. Please make sure you have all pages.
2. The point value of each problem occurs to the left of the problem.
Fall 2010
3. You must show correct work to receive credit. Correct answers with inconsistent
work or with no justification will not be given credit.
4, Books, notes and calculators are not allowed.
5. Turn off and put away all cell phones.
Page | Points | Points Possible
2 Uy 5
3 q 9
THT
5 U 5
6 | 4 5
Total | QG 30
Math 241 : Quiz 2 Fall 2010
Name: L brahim Talabl.
1. (3 pts) List, up to isomorphism, all abelian groups of order 600. Which of these groups
igevelic? TBath
Soy First, Ae find the prime factodzation of aie
Iso. \2
75 \3
28 5
= 600= Ae Fee” . 1 ;
So, any abelian giaup of olde/ 600 must be isomo(ehic fo
one ot the Gllowing.
~ 2¢ x L3axZas
- Z¢ y 23x26 xbZe
« Zor Zq x Ze xls (Z)
- 22 «Zy Ls «LZ6 «£5
> Z, x2," 2, xX 23% Zoe
a Zo. «Zo«Z, x Ze «Zegu Zo.
2. (2 pts) Find the order of (Z12 x Zig)/ < (10,12) >.
inde
-_ (Zi * Zig) /< (0,12)) =
(ZiexZ) < (10,2)
Math 241 Quiz 2 Fall 2010
Name Lb@him _Latabl,-.
3. (3 pts) Find the order of (4,2)+ < (4,4) > im (Zg x Zs)/ < (4,4) >.
BLU AAU nak
< C407 = F(49, (9,0), (0,4), 4,0),(4, 4)
(0,0) 7 ;
pote that AM4,2) = (24) E <(4,4)> (3)
50 8H,2) +<(4,4)>] BZ = <u>
hene, whe order 16: ae cling fo
4. (3 pts) Show that Zs x Zg is not isomorphic to Zig.
— oe a
as 3ad6 fil tb be ah plme na s,
Zi = Lq x Ze Since :
Z3 «Ze ~ Zs x Zz «Zo. (3)
but ~23xZe £ DZq as 343 weshelahvel
fide.
5. (3 pts) Find a nontrivial homomorphism ¢ : Zo4 > Zig.
onsider the homo marphi smn i) :
d(o)-° | QD is 0. hem -
bce (3h p zn re
f (3) =13 d is ono <4>
6(4) +0 oF Ze
1 38 So d is a hom
o() =(0 mod) 14 anote that [karl=4
vhee ‘ng Zo divides beth
\Zule2y %\Ze\= 16
Math 241 Quiz 2 Fall 2010
Name: T bihin Tatalole .
6. Let S3 be the group of permutations on 3 letters.
(a) (3 pts) Find the left cosets of < (1,2) > in $3.
a FOB
» PEL dn
i lSs
SS sing Ke] =2 | | loa) 2 2 £23
so thee ge = cesek see C(yy> in @
Sol 1, 2)(ia)= A < tales £1, 2)Y = (C12) fe} :
. (1,3) (2) = (4309 UN2F =f (1 23) 1,3)
- (2:3) ((LA> = fe 312), (e.323 ; i : ee
(b) (3 pts) Is < (1,2) > anormal subgroup of $3?
No, because <(1, 07.C1,3) = (693), R(3)f
63) = 7032) ) (13)§
+ (13 <a>
thee. is an elemear (ak leash one tr
BLY, Oo eongh lo prove
false}
iA S3 by which
a. LO, FD?
‘ (igh eft @seF ¢ righh oath
Math 241 Quiz 2
Name: brahin Tatobly.
Fall 2010
7. (5 pts) Let H be a subgroup of a group G and let N(H) = {g € G|gHg™' = H}. Show
that N(p)) is a subgroup of G.
NCH fg eG | gts =Hf
mic prove NCH)
c) closwe :
Let 9,75. € NCH)
ms pele 4,495 Ent) phe, ;
: WS Hg 4)" am
since g,tig - es pthen for h €Y,
@ how vloe 5h! & As some.
wW ie qh ‘ 4, 4, niet
vey invel(se¢ fet g é NCH) > 3"! exists A gbup &
> gig” =
=> ghg- sh’ for
sre h, hGH
a
Math 241 Quiz 2 Fall 2010
The hin Tatab \F
Name:
3. (5 pts) Let H be a subgroup of a group G. Show that gH is a subgroup of G if and only
ifg € H.
~ geH
Move gt se 3
(<=) fe suppose Ge"
CZ fag gt _ {9 | he Hf
) idearty ele se, Eg wey 2
Dre] , bE Le y
Wy yeness te Re ese