5310-6310-test-ch2-answer (1)

5310-6310-test-ch2-answer (1) - Abstract Algebra Chapter II...

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Unformatted text preview: Abstract Algebra Chapter II Name: A HS W 8!!" Totally 110 points, including 10 bonus ones. Please finish in 50 minutes. Here (3) Sn denotes the symmetric group of n letters; (b) The order of a finite group is the number of elements in thie group: (c) The order of a group element. is the order of the cyclic subgroup it generates. 56 d 123456 an 7: 41 513264 are two permutations in .95. Do the following exercises: 1. (48 points) Suppose that (a) Compute (77 (be aware of the multiplication orderl). _ 123%95Vi13 7‘; '_ dt”(glg;-/+:j\9r31é‘rl“f ; (b) Compute 0—1. a“: (3:2! W'l‘ 5“” — (h L, 1 fl (a (K'- r N _. w w, l‘ —+‘ W to ex v Math 5310—6310 Chapter 11 Test, Page '2 of 3 OeLober 13, 2006 (d) What. is the order of a in 86'? The “die” a? "’99 “‘5?!1HUSéQ‘UnS‘}?!:Lcm(3,z):5 (e) Recall that a cycle can be decomposed into products of transpositions: (a11a21' . ‘ fan) = (0113an)(a12an—1)"'(a’11(‘L3)(a‘lfla'2)' . . ‘7 ’ ' ' . i . bse this to decompose a = g g j" into producrs of transposuyions. (f) is a an odd permutation or an even permutation? 6 {,3 an odd pervmicité'm, 2. (1'2 points) Find all cosets of the subgroup 6Z of 2Z. Themsflfic’fléfl 622., 21’61’ Cmd ‘i—TéZ 3. (20 points) Suppose Lhat every G; is a group for i = 1, o -- .n. Prove that H; 0,: satisfies the assocaatnrity law (We need tins in proving that, 11:101- 18 a group). PTCC’F'V Lat Lethal) V"; an) {2%, in; ha), and LC: ,. Ca, “)Cw) I _ _ m _ be gm;- Jrlaree demerits w HER. ihm 1:.i K ’ ‘— ‘s'c c -~,C ) IKQQE.Q«2."JCW3(bubh‘ mm L h n ’ .. ‘ 7‘ f C .,-..‘(;H) :: (Ciibg.9\102,‘ ,.C\-.n13..)(. vs; 2., , rink“) Lfl) _ - ‘ J ‘. WC" “ - r’ ‘- 1 {ALfl/L' _7'} I K ' ' ‘ " ' . I bu {issuc‘mhv‘Jy 1m _ A - ck Hut ) L ' f :_ (CHL’DJu) , C'xxkhuh ’ n l n ) J Emir Weft/I} If '1 {GI-l,- aliu-j I'R}CE‘C‘I}71L2'} z‘ b" C”) : LCM/Q1 .: “7/ GU") Li‘bi-‘E’tan‘i bWM-C‘: CL) MUCH?) ' r“ ' . M I . r ‘50 £116.; Schs‘Fsec HQ cissocflmiudy tau/E Math 5310—6310 Chapter 11 Test, Page 3 of 3 October 1.3, 2006 4. (20 points) L-isL all abelian groups of order 72 up to isomorphism. ’71 : 23 x 52" By 'Thm H42 {F.IGQ-PJGQ) , W a»; {Harte aheh‘am group can be axpress as Z“? “I X x Z”, )rn when (in 7% (W PM“); | i} ‘ ‘ 23 @ Com 5% dtCOWPmEd RS ZXZX‘Z or 21x2, 0f 23 g?— Cam bfi CEELOmPoi'Ed 615 3X3 cr 751" SC: 5‘“ abeb‘am graufs order 7?. are ZLXZzXZszg‘AZ; 223 2‘: 23'!— 5. (10 points) What, is the order of the element (3,109) in 2;, x Zn x Z”? Tim: order cf 3 in 2;}. is an ‘r/gMUf‘r) 7* ‘1' TIM; Ord‘er of: 10 in 2,; 1:5 Iz/gcdézomz) :- 6 The orzié’r a? 01-19% '2‘; 1'35 1‘? /36d(9,35“) : g Them/:ch , @ bj Tkm H?! ((7.16?) } Hne order 01: (310,61) “in armada: is Lcml‘hbfi): 60 ...
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This note was uploaded on 10/12/2010 for the course MATH 5310 taught by Professor Staff during the Spring '08 term at Auburn University.

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5310-6310-test-ch2-answer (1) - Abstract Algebra Chapter II...

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