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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

## 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.

### Page1 / 3

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online