Contemporary Abstract Algebra
9th Edition
ISBN: 9781337249560
Author: Joseph Gallian
Publisher: Cengage Learning US
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Textbook Question
Chapter 4, Problem 60E
Given the fact that U(49) is cyclic and has 42 elements, deduce thenumber of generators that U(49) has without actually finding any ofthe generators.
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Find all generators of 60Z = {0,1,2,...,59}
Suppose α is a single cycle of length m and β is a single cycle of length n. What can be said about the order of the element αβ if α and β are disjoint cycles? If the cycles α and β share a single element in common? If the cycles α and β share two elements in common? (There are two different answers to this last question!)
b, c, a, or no generator, since it is not cyclic?
Chapter 4 Solutions
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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- Also compute η2arrow_forwardHow can I prove that Z4 is cylic? I know that It has something to do with generators (I think?) and the generators of Z4 are 1 and 3 (I don't understand why tho). Can someone please help? Thank you!!arrow_forwardIn general, if (12 · . · k) is a cycle of length k and o is any permutation then o(12 · · k)o -1 = (0(1)o(2). · · o(k))arrow_forward
- Suppose a permutation s swaps the first two elements of {1, 2, ..., n}. A permutation r rotates the elements: r(1)=2, r(2)=3, ... r(n-1)=n, r(n)=1. We can thus write in cycle notation: s=(1 2), r=(1 2 3 4 ... n). What does the permutation rm sr -m do?arrow_forwardCompute the following (1+ iv3)5arrow_forwardLet H = {(1), (12)(34), (13)(24), (14)(23)}. Find the left cosets of H in A4. How many left cosets of H in S4 are there? (Detemine this without listing them).arrow_forward
- The generators of Z12 are: O {1,5,7,8,11} O {1,5,7,11} O {1,3,7,11} O {1,3,5,7,11}arrow_forwardShow that U22 is cyclic. Find all of the generators of U22 and explain how you know that each element is indeed a generator.arrow_forward14. Consider the permutations a = (5 21 4 3), B = (1 3)(2 4 6), y = (12 4)(3 5 6) of {1,2, 3, 4, 5, 6} expressed in cycle notation. Which one of the following is correct? (a) a and y are odd, and B is even. (b) a and y are even, and ß is odd. (c) a and B are even, and y is odd. (d) a and B are odd, and y is even. (e) B and y are even, and a is odd.arrow_forward
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