MTsols-math120

# MTsols-math120 - Practice Midterm Math 120A October 2009...

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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! {Lt-6d 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-—( v-l 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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