Elements Of Modern Algebra

8th Edition

ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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9. Suppose that and are subgroups of the abelian group such that . Prove that .

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Exercises
3. Find an isomorphism from the additive group to the multiplicative group of units .
Sec.
16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.

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5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:

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Prove or disprove that H={ [ 1a01 ]|a } is a normal subgroup of the special linear group SL(2,).

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let Un be the group of units as described in Exercise16. Prove that [ a ]Un if and only if a and n are relatively prime. Exercise16 For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication.

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Exercises
31. Let be a group with its center:
.
Prove that if is the only element of order in , then .

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Prove that each of the following subsets H of GL(2,C) is subgroup of the group GL(2,C), the general linear group of order 2 over C a. H={ [ 1001 ],[ 1001 ],[ 1001 ],[ 1001 ] } b. H={ [ 1001 ],[ i00i ],[ i00i ],[ 1001 ] }

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Exercises
18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .

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Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.

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Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under addition. Find H1+H2 and determine if the sum is direct.

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11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.

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13. Assume that are subgroups of the abelian group . Prove that if and only if is generated by

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