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10.2.11. Let G = D34 = (a,b | a¹7 = b² = e,ba = a¹6b). Let H = (a) and K = = (b). (a) Draw a partial lattice diagram of subgroups of G that includes the subgroups H and K. Include also H¦ K and (H,K). Label the edges with appropriate numbers, and give reasons for what you have done. In what follows, you may refer back to your diagram. (b) Is H

10.2.11. Let G = D34 = (a,b | a¹7 = b² = e,ba = a¹6b). Let H = (a) and K = = (b). (a) Draw a partial lattice diagram of subgroups of G that includes the subgroups H and K. Include also H¦ K and (H,K). Label the edges with appropriate numbers, and give reasons for what you have done. In what follows, you may refer back to your diagram. (b) Is H

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.4: Cosets Of A Subgroup
Problem 12E: Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order...
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10.2.11. Let G = D34 = (a,b | a¹7 = b² = e,ba = a¹6b). Let H = (a) and K = = (b). (a) Draw a partial lattice diagram of subgroups of G that includes the subgroups H and K. Include also H¦ K and (H,K). Label the edges with appropriate numbers, and give reasons for what you have done. In what follows, you may refer back to your diagram. (b) Is H<G? What is NG(H)? What is NG(K)? Why? Is K <G? (c) What is clG(a)|? What is clG(b)|? Why? Note that clG (x) is the conjugacy class of x in G. (d) What is Z(G)? Why?
Transcribed Image Text:10.2.11. Let G = D34 = (a,b | a¹7 = b² = e,ba = a¹6b). Let H = (a) and K = = (b). (a) Draw a partial lattice diagram of subgroups of G that includes the subgroups H and K. Include also H¦ K and (H,K). Label the edges with appropriate numbers, and give reasons for what you have done. In what follows, you may refer back to your diagram. (b) Is H<G? What is NG(H)? What is NG(K)? Why? Is K <G? (c) What is clG(a)|? What is clG(b)|? Why? Note that clG (x) is the conjugacy class of x in G. (d) What is Z(G)? Why?
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