Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Chapter 7, Problem 48E
To determine
Give the explanation for the given proof.
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Contemporary Abstract Algebra
Ch. 7 - Let H=0,3,6,9,... . Find all the left cosets of H...Ch. 7 - Rewrite the condition a1bH given in property 6 of...Ch. 7 - Let n be a positive integer. Let H=0,n,2n,3n,... ....Ch. 7 - Find all of the left cosets of {1, 11} in U(30).Ch. 7 - Suppose that a has order 15. Find all of the left...Ch. 7 - Let a andb be elements of a group G and H and K be...Ch. 7 - If H and K are subgroups of G and g belongs to G,...Ch. 7 - Suppose that K is a proper subgroup of H and H is...Ch. 7 - Let G be a group with G=pq , where p and q are...Ch. 7 - Suppose H and K are subgroups of a group G. If...
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- 45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )arrow_forwardLet G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.arrow_forwardIf a is an element of order m in a group G and ak=e, prove that m divides k.arrow_forward
- 18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.arrow_forwardLet be a subgroup of a group with . Prove that if and only if .arrow_forwardLet be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.arrow_forward
- If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.arrow_forwardExercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.arrow_forwardLet H1 and H2 be cyclic subgroups of the abelian group G, where H1H2=0. Prove that H1H2 is cyclic if and only if H1 and H2 are relatively prime.arrow_forward
- 1. Consider , the groups of units in under multiplication. For each of the following subgroups in , partition into left cosets of , and state the index of in a. b.arrow_forward9. Find all homomorphic images of the octic group.arrow_forwardExercises 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_forward
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