Let H = {β ∈ S5 : β(4) = 4}. Prove that H is a subgroup of S5. (Reminder: The group operation of S5 is composition of functions, and it’s helpful here to use the definition (αβ)(n) = α(β(n)).)
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Let H = {β ∈ S5 : β(4) = 4}. Prove that H is a subgroup of S5.
(Reminder: The group operation of S5 is composition of functions, and it’s helpful here to use the definition (αβ)(n) = α(β(n)).)
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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 ] }Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=4014. Let be an abelian group of order where and are relatively prime. If and , prove that .
- 15. Prove that each of the following subsets of is subgroup of the group ,the general linear group of order over. a. b. c. d.Let 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.5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:
- Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.In D12 = <x, y | x2 = y12 = e, xyx = y-1>, prove that the subgroupH = <x, y3> (which is isomorphic to D4) is not a normal subgroup.You have previously proved that the intersection of two subgroups of a group G is always a subgroup. For G = S3, show that the union of two subgroups may not be a subgroup by providing a counterexample.
- Find the subgroup of Dn generated by r2 and r2s, distinguishing carefully between the cases n odd and n even.How would you solve such a question: Suppose f : Z → Z5 is a group homomorphism, and suppose f (3) = 2.Find f (1).What can you say about a subgroup of D3 that contains R240 and areflection F? What can you say about a subgroup of D3 that containstwo reflections?