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Unformatted text preview: Practice Midterm
Math 120A
October, 2009 Instructions: You have 50 minutes to complete this exam. All relevant
work must be shown to receive full credit for a problem. No credit will
be given for illegible solutions. Problerri’ 1: (a): (7 points) Recall from lecture that an operation of a group G
on a set S is a function G X S —> S satisfying
(1) as = 3 VS 6 S
(2) g(hs) : (gh)s Vg,h E G, s E S
If s E S , show that the stabilizer of 3, deﬁned as the set {96 Glgs= 8}
is a subgroup of G' 3 a667 65:5 Wm C‘ ill Closure under brim?! 0f (a since A525
It 5:5) h5£$ Marx ~ ~ I
~ ( MS“: (ks): g = 3
S0 3mg: 5 {e doggy! a j i ”(Sr/Ice :3
Nil C(csoza order invarses 13760 lids, @166 5:55 $— 6. f 5? (b): (3 oints Why does the set { 1, 23} of nonzero classes mod
ulo 4 fail to be a group under addition? Under multiplication? Mﬂm No arm/e (ole/WI (sic/Ia? La Ol
Nat closey’U/vz 6r Ham Hultiflz'CatIm‘ i has l‘o I’m/arse
[also
Malawi WI 93l~wf S“l who)
I‘ ll N\O\ 1N} N Problem 2: (a): (6 points) Let G be a. group with at least two elements and no
proper nontrivial subgroups. Show that [GI is a prime number. fkl 66’ ‘feﬁlm <7i5amni‘rivml ‘ r ,SOESM ate. G=<>
g is (1 (MC X Alamnvidemﬁk :j’re enera’rgf . g
3?; 6 were ilex'nﬁﬂ) «Hen 3 :50 kism ¢ 3 "’—7 6, 61ml ¢ fakes Waters
{a gated $ M: f’. Titus} 93 is not at aerator of 6/, SiAce Xis Ora at”!
S ‘ 55G} 7&8 (since ¢"C€\:03 ﬁlis“ , bits all romdem‘iyiwg 6 Z‘
. t er
1103.: GL5 ﬁniﬁeC 71.6956; (Gl=/\/d*\9( 6/45 [M » gate” @155.
g ' p 2‘6 _ ‘ 06 (allel‘fﬂﬂﬁs :ﬁ‘eﬁi
Wale/)7; 6 Las \{aezzl teaénrqmztiﬂmm 50 6M ll f" 0 ' z x
”(735% gr ﬁfe/EM [3,. ail/6F Lzaéfjm/ri—lan: reézfA/défnw i
l I (b): (4 points) In the group (Zia of nonzero classes modulo 13 under
multiplication, ﬁnd the subgroup generated by § and E. N016! & T62?
45:50? :13? as?__ —— —l J ' sage}? =§CI
dbd 16'?:_:§'g4:r éﬁ/ 2'3“” 3/WAENCQr/€= ~
M if"??? m mew w a tip/{fa in :T'j Problem 3: (a): (4 points) Write the permutation _1234567
”“2517346 as a product of disjoint cycles in 87. 0V: (L15,;§5)(4ﬂ,6) (b): (3 points) What is the order of a? w SEE 0‘: 032.2% '' d F06 orJEF 4}
ﬁ: @9ij " (5 A45 order ’5 / Egg Sediitﬁi o< M50I21€P¢l5 has Oldef‘g/ amt (cmélgaﬂzﬂj So 0' {as ad€r$&\ W... (c): (3 points) Is 0 even or odd? “7— ( 5510, 91 0:30
E: (Lejﬂé‘t,?\ so 034?:(£12WC[)§WCIJX\CLI‘JC\CLE?\ WV
S’tfal S‘oaiganCL Problem 4: Let H be the subset of M2 (R) consisting of all matrices of the form [:1 if] for a, b E R You showed in your homework that H is closed under matrix multiplication. (a): (7 points) Show that (C ) is isomorphic to (H, '.>
'1 Oeime. 925 «3—41 )7 756*)“: [6; fl ¢l§mi0 q—i canes {mm 4+5)! ¢is (1: [a in  {C 41:36“: C [Kd SO @Yatbil— ¢CCWH 3a+5i=ctdl
Ham? + : (ﬁbi’fad‘ll ac gamma
“F7 ﬂzim—M QEC M0! CC (.4) a 49 C47!
{Lt6d 61(3th )2 [bald ac:b¢£]:[b &][7( C}
(b): (3 points) Find the order of the matrix [i —3 in H 2% Aﬂé‘jﬂli)
l ”I
Set M3 [t 1]
77am“ L—( vl c772"
(“1&3th OJ
[0 "2][67 ”l 5'? 0—\
LG $30]:[ Jf : 4‘]:
50 (1‘5: WM)? (61
NW: (WY: (WI
5m n‘jnf’n‘f W“ ale «((4,511,in J <H> 05 tit/ale 3) M {as {Narnia omkr Lt_
”a ...
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 Spring '09
 SMITH
 Math

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