Contemporary Abstract Algebra
9th Edition
ISBN: 9781337249560
Author: Joseph Gallian
Publisher: Cengage Learning US
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Textbook Question
Chapter 4, Problem 38E
Let m and n be elements of the group Z. Find a generator for thegroup
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Contemporary Abstract Algebra
Ch. 4 - Find all generators of Z6,Z8,andZ20 .Ch. 4 - Suppose that a,b,andc are cyclic groups of orders...Ch. 4 - List the elements of the subgroups 20and10inZ30 ....Ch. 4 - List the elements of the subgroups 3and15inZ18 ....Ch. 4 - List the elements of the subgroups 3and7inU(20) .Ch. 4 - What do Exercises 3, 4, and 5 have in common? Try...Ch. 4 - Find an example of a noncyclic group, all of whose...Ch. 4 - Let a be an element of a group and let a=15 ....Ch. 4 - Prob. 9ECh. 4 - In Z24 , list all generators for the subgroup of...
Ch. 4 - Let G be a group and let aG . Prove that a1=a .Ch. 4 - In Z, find all generators of the subgroup 3 . If a...Ch. 4 - In Z24 , find a generator for 2110 . Suppose that...Ch. 4 - Suppose that a cyclic group G has exactly three...Ch. 4 - Let G be an Abelian group and let H=gG||g divides...Ch. 4 - Complete the statement: a|=|a2 if and only if |a|...Ch. 4 - Complete the statement: a2|=|a12 if and only if ....Ch. 4 - Let a be a group element and a= . Complete the...Ch. 4 - If a cyclic group has an element of infinite...Ch. 4 - Suppose that G is an Abelian group of order 35 and...Ch. 4 - Let G be a group and let a be an element of G. a....Ch. 4 - Prove that a group of order 3 must be cyclic.Ch. 4 - Let Z denote the group of integers under addition....Ch. 4 - For any element a in any group G, prove that a is...Ch. 4 - If d is a positive integer, d2 , and d divides n,...Ch. 4 - Find all generators of Z. Let a be a group element...Ch. 4 - Prove that C*, the group of nonzero complex...Ch. 4 - Let a be a group element that has infinite order....Ch. 4 - List all the elements of order 8 in Z8000000 . How...Ch. 4 - Suppose that G is a group with more than one...Ch. 4 - Let G be a finite group. Show that there exists a...Ch. 4 - Determine the subgroup lattice for Z12 ....Ch. 4 - Determine the subgroup lattice for Z8 . Generalize...Ch. 4 - Prove that a finite group is the union of proper...Ch. 4 - Show that the group of positive rational numbers...Ch. 4 - Consider the set {4, 8, 12, 16}. Show that this...Ch. 4 - Give an example of a group that has exactly 6...Ch. 4 - Let m and n be elements of the group Z. Find a...Ch. 4 - Suppose that a andb are group elements that...Ch. 4 - Prob. 40ECh. 4 - Prob. 41ECh. 4 - Let F and F’be distinct reflections in D21 . What...Ch. 4 - Suppose that H is a subgroup of a group G and H=10...Ch. 4 - Prob. 44ECh. 4 - If G is an infinite group, what can you say about...Ch. 4 - If G is a cyclic group of order n, prove that for...Ch. 4 - For each positive integer n, prove that C*, the...Ch. 4 - Prove or disprove that H=nZn is divisible by both...Ch. 4 - Prob. 49ECh. 4 - Prob. 50ECh. 4 - Prob. 51ECh. 4 - Prob. 52ECh. 4 - Prob. 53ECh. 4 - Prob. 54ECh. 4 - Prob. 55ECh. 4 - Prob. 56ECh. 4 - Prob. 57ECh. 4 - Prob. 58ECh. 4 - Prove that no group can have exactly two elements...Ch. 4 - Given the fact that U(49) is cyclic and has 42...Ch. 4 - Let a andb be elements of a group. If a=10andb=21...Ch. 4 - Let a andb belong to a group. If |a| and |b| are...Ch. 4 - Let a andb belong to a group. If a=24andb=10 ,...Ch. 4 - Prove that U(2n)(n3) is not cyclic.Ch. 4 - Prove that for any prime p and positive integer...Ch. 4 - Prove that Zn has an even number of generators if...Ch. 4 - If a5=12 , what are the possibilities for |a|? If...Ch. 4 - Suppose that x=n . Find a necessary and sufficient...Ch. 4 - Let a be a group element such that a=48 . For each...Ch. 4 - Prove that H={[1n01]|nZ} is a cyclic subgroup of...Ch. 4 - Suppose that |a| and |b| are elements of a group...Ch. 4 - Let a andb belong to a group. If a=12,b=22,andabe...Ch. 4 - Determine (81),(60)and(105) where is the Euler...Ch. 4 - If n is an even integer prove that (2n)=2(n) .Ch. 4 - Let a andb belong to some group. Suppose that...Ch. 4 - For every integer n greater than 2, prove that the...Ch. 4 - (2008 GRE Practice Exam) If x is an element of a...
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- Use mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.arrow_forwardLet a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and d1.arrow_forwardProve that Ca=Ca1, where Ca is the centralizer of a in the group G.arrow_forward
- Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.arrow_forward38. Let be the set of all matrices in that have the form with all three numbers , , and nonzero. Prove or disprove that is a group with respect to multiplication.arrow_forwardUse mathematical induction to prove that if a1,a2,...,an are elements of a group G, then (a1a2...an)1=an1an11...a21a11. (This is the general form of the reverse order law for inverses.)arrow_forward
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- Prove or disprove that the set of all diagonal matrices in Mn() forms a group with respect to addition.arrow_forward39. Let be the set of all matrices in that have the form for arbitrary real numbers , , and . Prove or disprove that is a group with respect to multiplication.arrow_forward20. Let and be elements of a group . Use mathematical induction to prove each of the following statements for all positive integers . a. If the operation is multiplication, then . b. If the operation is addition and is abelian , then .arrow_forward
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