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Elements Of Modern Algebra
- 9. Suppose that and are subgroups of the abelian group such that . Prove that .arrow_forward11. 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.arrow_forward13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forward
- 2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .arrow_forwardLet G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.arrow_forwardProve or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.arrow_forward
- Find two groups of order 6 that are not isomorphic.arrow_forwardProve 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 ] }arrow_forwardSuppose 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.arrow_forward
- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.arrow_forwardExercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .arrow_forwardConsider the group U9 of all units in 9. Given that U9 is a cyclic group under multiplication, find all subgroups of U9.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,