Contemporary Abstract Algebra
9th Edition
ISBN: 9781337249560
Author: Joseph Gallian
Publisher: Cengage Learning US
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Chapter 3, Problem 23E
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Chapter 3 Solutions
Contemporary Abstract Algebra
Ch. 3 - For each group in the following list, find the...Ch. 3 - Let Q be the group of rational numbers under...Ch. 3 - Let Q and Q* be as in Exercise 2. Find the order...Ch. 3 - Prove that in any group, an element and its...Ch. 3 - Without actually computing the orders, explain why...Ch. 3 - In the group Z12 , find a,b,anda+b for each case....Ch. 3 - If a, b, and c are group elements and a=6,b=7 ,...Ch. 3 - What can you say about a subgroup of D3 that...Ch. 3 - What can you say about a subgroup of D4 that...Ch. 3 - How many subgroups of order 4 does D4 have?
Ch. 3 - Determine all elements of finite order in R*, the...Ch. 3 - Complete the statement “A group element x is its...Ch. 3 - For any group elements a and x, prove that xax1=a...Ch. 3 - Prove that if a is the only element of order 2 in...Ch. 3 - (1969 Putnam Competition) Prove that no group is...Ch. 3 - Let G be the group of symmetries of a circle and R...Ch. 3 - For each divisor k1 of n, let Uk(n)=xU(n)xmodk=1...Ch. 3 - Suppose that a is a group element and a6=e . What...Ch. 3 - If a is a group element and a has infinite order,...Ch. 3 - For any group elements a and b, prove that ab=ba .Ch. 3 - Show that if a is an element of a group G, then...Ch. 3 - Show that U(14)=3=5 . [Hence, U(14) is cyclic.] Is...Ch. 3 - Show that U(20)k for any k in U(20). [Hence, U(20)...Ch. 3 - Suppose n is an even positive integer and H is a...Ch. 3 - Let n be a positive even integer and let H be a...Ch. 3 - Prove that for every subgroup of Dn , either every...Ch. 3 - Let H be a subgroup of Dn of odd order. Prove that...Ch. 3 - Prove that a group with two elements of order 2...Ch. 3 - Prob. 29ECh. 3 - Prob. 30ECh. 3 - Prob. 31ECh. 3 - Suppose that H is a subgroup of Z under addition...Ch. 3 - Prove that the dihedral group of order 6 does not...Ch. 3 - If H and K are subgroups of G, show that HK is a...Ch. 3 - Let G be a group. Show that Z(G)=aGC(a) . [This...Ch. 3 - Let G be a group, and let aG . Prove that...Ch. 3 - For any group element a and any integer k, show...Ch. 3 - Let G be an Abelian group and H=xG||x is odd}....Ch. 3 - Prob. 39ECh. 3 - Prob. 40ECh. 3 - Let Sbe a subset of a group and let H be the...Ch. 3 - In the group Z, find a. 8,14 ; b. 8,13 ; c. 6,15 ;...Ch. 3 - Prove Theorem 3.6. Theorem 3.6 C(a) Is a Subgroup...Ch. 3 - If H is a subgroup of G, then by the centralizer...Ch. 3 - Must the centralizer of an element of a group be...Ch. 3 - Suppose a belongs to a group and a=5 . Prove that...Ch. 3 - Prob. 47ECh. 3 - In each case, find elements a and b from a group...Ch. 3 - Prove that a group of even order must have an odd...Ch. 3 - Consider the elements A=[0110]andB=[0111] from...Ch. 3 - Prob. 51ECh. 3 - Give an example of elements a and b from a group...Ch. 3 - Consider the element A=[1101] in SL(2,R) . What is...Ch. 3 - For any positive integer n and any angle , show...Ch. 3 - Prob. 55ECh. 3 - In the group R* find elements a and b such that...Ch. 3 - Prob. 57ECh. 3 - Prob. 58ECh. 3 - Prob. 59ECh. 3 - Compute the orders of the following groups. a....Ch. 3 - Let R* be the group of nonzero real numbers under...Ch. 3 - Compute U(4),U(10),andU(40) . Do these groups...Ch. 3 - Find a noncyclic subgroup of order 4 in U(40).Ch. 3 - Prove that a group of even order must have an...Ch. 3 - Let G={[abcd]|a,b,c,dZ} under addition. Let...Ch. 3 - Let H=AGL(2,R)detA is an integer power of 2}. Show...Ch. 3 - Let H be a subgroup of R under addition. Let...Ch. 3 - Let G be a group of functions from R to R*, where...Ch. 3 - Let G=GL(2,R) and...Ch. 3 - Let H=a+bia,bR,ab0 . Prove or disprove that H is...Ch. 3 - Let H=a+bia,bR,a2+b2=1 . Prove or disprove that H...Ch. 3 - Let G be a finite Abelian group and let a and b...Ch. 3 - Prob. 73ECh. 3 - If H and K are nontrivial subgroups of the...Ch. 3 - Prob. 75ECh. 3 - Prove that a group of order n greater than 2...Ch. 3 - Let a belong to a group and a=m. If n is...Ch. 3 - Let G be a finite group with more than one...
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- Exercises 13. For the given permutations, and , find a permutation such that is the conjugate of by –that is, such that . a. ; b. ; c. ; d. ; e. ; f. ;arrow_forwardLet be as described in the proof of Theorem. Give a specific example of a positive element of .arrow_forwardProve the following: a) Show that ||ủ + ở|| < ||ū|| + ||3||. b) Show that ||ū x ³||² = (ū · ū)( · i) – (ū · 3)².arrow_forward
- d) Draw an arrow diagram of the inverse for the bijection function found in (c). Define W:Z → Z and V: Z → Z by the rules W(a) = a? and V (a) = a mod 5 for all integers a. Find the following: e) (W • V)(6) (V • W)(9) iii. i. ii. Is (W • V) = (V •W) for all integers a e Z? Justify your answer. -1-arrow_forwardProve that A u (B n C) = (Au B) n (A u C).arrow_forwardLet A = {1, 2,...,n} and consider the functions f: A → A. a. How many functions satisfy f(1) #1? b. How many functions satisfy f(1) #1 or f(2) ‡ 2? c. Of the functions f: A → A with f(1) = 2 and f(2)= 3, how many are bijections?arrow_forward
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