8th Edition

ISBN: 9781285463261

Author: Douglas Smith, Maurice Eggen, Richard St. Andre

Publisher: Cengage Learning

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Chapter 3.1, Problem 6E

Let ( G , ∘ ) and ( H , * ) be groups, i be the identity element for H, and h : ( G , o ) → ( H , * ) be a homomorphism. The kernel of f is ker ( f ) = { x ∈ G : f ( x ) = i } . Thus, ker(f) is all the elements of G that map to the identity in H. Show that ker(f) is a subgroup of G.

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Chapter 3.1, Problem 5E

Chapter 3.1, Problem 7E

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Chapter 3 Solutions

A Transition to Advanced Mathematics

Ch. 3.1 - Let 3 and 6 be the sets of integer multiples of 3...Ch. 3.1 - Let (3,+) and (6,+) be the groups in Exercise 10,...Ch. 3.1 - Let ({a,b,c},o) be the group with the operation...Ch. 3.1 - (a)Prove that the function f:1824 given by f(x)=4x...Ch. 3.1 - Define f:1512 by f(x)=4x. Prove that f is a...Ch. 3.1 - Let (G,) and (H,*) be groups, i be the identity...Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prove that the relation of isomorphism is an...Ch. 3.1 - Prob. 10E

Ch. 3.1 - Prove that if G is a group and H is a subgroup of...Ch. 3.1 - Prob. 12E Ch. 3.1 - Prob. 13E Ch. 3.1 - Prob. 14E Ch. 3.1 - Prob. 15E Ch. 3.1 - Prob. 16E Ch. 3.1 - Prob. 17E Ch. 3.2 - (a)Show that any two groups of order 2 are...Ch. 3.2 - (a)Show that the function h: defined by h(x)=3x is...Ch. 3.2 - Let R be the equivalence relation on ({0}) given...Ch. 3.2 - Let (R,+,) be an integral domain. Prove that 0 has...Ch. 3.2 - Complete the proof of Theorem 6.5.5. That is,...Ch. 3.2 - Prob. 6E Ch. 3.2 - Assign a grade of A (correct), C (partially...Ch. 3.2 - Prob. 8E Ch. 3.2 - Prob. 9E Ch. 3.2 - Use the method of proof of Cayley's Theorem to...Ch. 3.2 - Prob. 11E Ch. 3.2 - Assign a grade of A (correct), C (partially...Ch. 3.2 - Prob. 13E Ch. 3.2 - Define on by setting (a,b)(c,d)=(acbd,ad+bc)....Ch. 3.2 - Prob. 15E Ch. 3.2 - Let f:(A,)(B,*) and g:(B,*)(C,X) be OP maps. Prove...Ch. 3.2 - Prob. 17E Ch. 3.2 - Let Conj: be the conjugate mapping for complex...Ch. 3.2 - Prove the remaining parts of Theorem 6.4.1.Ch. 3.3 - Let 3={3k:k}. Apply the Subring Test (Exercise...Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Prob. 6E Ch. 3.3 - Use the definition of “divides” to explain (a) why...Ch. 3.3 - Prob. 8E Ch. 3.3 - Prob. 9E Ch. 3.3 - Complete the proof that for every m,(m+,) is a...Ch. 3.3 - Define addition and multiplication on the set ...Ch. 3.3 - Prob. 12E Ch. 3.3 - Let (R,+,) be a ring and a,b,R. Prove that b+(a)...Ch. 3.3 - Prove the remaining parts of Theorem 6.5.3: For...Ch. 3.3 - We define a subring of a ring in the same way we...Ch. 3.4 - Prob. 1E Ch. 3.4 - Prob. 2E Ch. 3.4 - If possible, give an example of a set A such that...Ch. 3.4 - Let A. Prove that if sup(A) exists, then...Ch. 3.4 - Let A and B be subsets of . Prove that if sup(A)...Ch. 3.4 - a.Give an example of sets A and B of real numbers...Ch. 3.4 - a.Give an example of sets A and B of real numbers...Ch. 3.4 - An alternate version of the Archimedean Principle...Ch. 3.4 - Prob. 9E Ch. 3.4 - Prob. 10E Ch. 3.4 - Prob. 11E Ch. 3.4 - Prob. 12E Ch. 3.5 - Prob. 1E Ch. 3.5 - Prob. 2E Ch. 3.5 - Let A be a subset of . Prove that the set of all...Ch. 3.5 - Prob. 4E Ch. 3.5 - Let be an associative operation on nonempty set A...Ch. 3.5 - Suppose that (A,*) is an algebraic system and * is...Ch. 3.5 - Let (A,o) be an algebra structure. An element lA...Ch. 3.5 - Let G be a group. Prove that if a2=e for all aG,...Ch. 3.5 - Give an example of an algebraic structure of order...Ch. 3.5 - Prove that an ordered field F is complete iff...Ch. 3.5 - Prove that every irrational number is "missing"...Ch. 3.5 - Find two upper bounds (if any exits) for each of...Ch. 3.5 - Prob. 13E Ch. 3.5 - Prob. 14E Ch. 3.5 - Prob. 15E Ch. 3.5 - Let A and B be subsets of . Prove that if A is...Ch. 3.5 - Prob. 17E Ch. 3.5 - Prob. 18E Ch. 3.5 - Give an example of a set A for which both A and Ac...Ch. 3.5 - Prob. 20E Ch. 3.5 - Prob. 21E Ch. 3.5 - Prob. 22E

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Let ( G , ∘ ) and ( H , * ) be groups, i be the identity element for H, and h : ( G , o ) → ( H , * ) be a homomorphism. The kernel of f is ker ( f ) = { x ∈ G : f ( x ) = i } . Thus, ker(f) is all the elements of G that map to the identity in H. Show that ker(f) is a subgroup of G.

Solution Summary: The author explains the inverse of the function R_1=left (x,y)in Rtimes R:y=xright.

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Chapter 3 Solutions