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# ABSTRACT ALGEBRA 1 MATH 3140

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This 2 page Study Guide was uploaded by Cydney Conroy on Thursday October 29, 2015. The Study Guide belongs to MATH 3140 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 32 views. For similar materials see /class/231827/math-3140-university-of-colorado-at-boulder in Mathematics (M) at University of Colorado at Boulder.

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Date Created: 10/29/15

Abstract Algebra 1 MATH 3140 HANDOUT 5 October 27 2008 Review for Exam 2 The material covered by Exam 2 can be found in Sections 4711 and 13715 Some questions on the exam may ask you to state de nitions or theorems Practice Problems 1 For each property below nd all a 6 S5 that satisfy the given property a 2 412 5 1o3 4 5 1 3 5 b a has order 6 C 02 123 4 5 Equot Show that if every nonidentity element of a group G has order 2 then G is abelian Hint Use the assumption for abab E G 9 Let G be a group of order 6 a Prove that if G is abelian then G Z6 b Now assume that G is nonabelian and show that i G has an element a of order 3 Hint Use Problem 2 ii N a is a normal subgroup of G Let b E G be such that b N iii every element of G is of the form aibj 0 S i S 2 0 S j S 1 and this form is unique Hint Think of the cosets of N iv half1 12 that is ba azb v b has order 2 Hint First nd the order of bN in GN G g 83 F Prove that if a group G of order gt 1 has no nontrivial proper subgroups then G E Z for some prime p Hint Argue rst that G is cyclic 9quot Let U be one of the elements of order 4 in D4 a Show that N is a normal subgroup of D4 oforder 2 if and only if N 02 b Show that every nonidentity element of D402 has order 2 C Use the Homomorphism Theorem to prove that D4 has no homomorphism onto Z4 Sa TRUE or FALSE Justify your answer a fa 6 Sn has odd order then a is an even permutation b There epists a nontrivial homomorphism Z10 a Z21 0 Any two groups of order 29 are isomorphic d IfG is a nite group and H is a nonempty subset of G such that H is closed under multiplication then H is a subgroup of G 1 Abstract Algebra 1 MATH 3140 HANDOUT 3 September 22 2008 Review for Exam 1 The material covered by Exam 1 can be found in Sections 073 and Handouts 172 Some questions on the exam may ask you to state de nitions or theorems Practice Problems H Equot 03 F 9quot Sa 7 OD For a set A let 73A be the set of all subsets of A called the power set of A Show that if A and B are sets such that 73A Q 73B7 then A Q B For each subset of R gtlt R below decide whether or not it determines a function R a R If it does7 is the function one to one ls it onto a oab RgtltRlbzia217 bgo b Tab RgtltRlbzia2717 bZO Let f A a B be an arbitrary function Prove that a if f is one to one and g h C a A are such that fog foh7 then 9 h b if f is not one to one7 then there exist functions 97 h Z2 a A such that fogfohandg7 h Show that if f A a B is a bijection and g B a A is a function such that fog i3 then 9 f l Let N be the relation on the set R4r gtlt R4r de ned by 11702 N 51752 a Prove that N is an equivalence relation b Visualizing the elements of R gtlt R4r as points in the rst quadrant7 give a geometric description of the partition corresponding to N c Find a function with domain R4r gtlt R4r whose kernel is N if and only if 1le agbl Let 7 s E R7 and de ne an operation gtk on R by mky rzsy for all my 6 R For which 7 s E R is the operation gtk a commutative7 b associative Show that the function p R7r a R27r de ned by a 2x for all z E R is an isomorphism of Rh 7r with R27H27r TRUE or FALSE Justify your answer a Iffg A a A are functions such that f 09 m then f 9 1 b If g A a A are functions such that f o g m then 9 is onc to onc c 57 S gtk if there mists a function p S a S that is not an iso morphism bctwccn S7 and 87 1

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