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Elements Of Modern Algebra
- Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forward5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:arrow_forwardLet be a subgroup of a group with . Prove that if and only if .arrow_forward
- 27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .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_forwardLet G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.arrow_forward
- Let 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_forwardLet G be the group and H the subgroup given in each of the following exercises of Section 4.4. In each case, is H normal in G? Exercise 3 b. Exercise 4 c. Exercise 5 d. Exercise 6 e. Exercise 7 f. Exercise 8 Section 4.4 Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup (1),(2,3) of S3. Find the distinct left cosets of H in S3, write out their elements, partition S3 into left cosets of H, and give [S3:H]. Find the distinct right cosets of H in S3, write out their elements, and partition S3 into right cosets of H. In Exercises 7 and 8, let G be the multiplicative group of permutation matrices I3,P3,P32,P1,P4,P2 in Example 6 of Section 3.5 Let H be the subgroup of G given by H=I3,P4={ (100010001),(001010100) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H. Let H be the subgroup of G given by H=I3,P3,P32={ (100010001),(010001100),(001100010) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H.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_forward
- Let 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_forwardLet G be a group and gG. Prove that if H is a Sylow p-group of G, then so is gHg1arrow_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,