Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Chapter 3, Problem 3E
Let Q and Q* be as in Exercise 2. Find the order of each element in Q and in Q*.
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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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- ma Classes Term. Spring 2025 Title Details Credit Hours CRN Schedule Type Grade Mode Level Date Status Message *MATHEMATICS FOR MANAGEME... MTH 245, 400 4 54835 Online Normal Grading Mode Ecampus Undergradu... 03/21/2025 Registered **Web Registered... *SOIL SCIENCE CSS 205, 400 0 52298 Online Normal Grading Mode Undergraduate 03/21/2025 Waitlisted Waitlist03/21/2025 PLANT PATHOLOGY BOT 451, 400 4 56960 Online Normal Grading Mode Undergraduate 03/21/2025 Registered **Web Registered... Records: 3 Schedule Schedule Detailsarrow_forwardHere is an augmented matrix for a system of equations (three equations and three variables). Let the variables used be x, y, and z: 1 2 4 6 0 1 -1 3 0 0 1 4 Note: that this matrix is already in row echelon form. Your goal is to use this row echelon form to revert back to the equations that this represents, and then to ultimately solve the system of equations by finding x, y and z. Input your answer as a coordinate point: (x,y,z) with no spaces.arrow_forward1 3 -4 In the following matrix perform the operation 2R1 + R2 → R2. -2 -1 6 After you have completed this, what numeric value is in the a22 position?arrow_forward
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