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Let G = D10 × Z/7Z, where, as usual, D10 = (a, b | a³ = b² = e, ba = a*b). Let H ((b, 1)) and K = D10 × {0}. (a) Draw a partial lattice diagram of subgroups of G that includes the subgroups H and K. Include also H N 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) What is NG(H), the normalizer of H in G? Why? Is NG(K) = G?

Let G = D10 × Z/7Z, where, as usual, D10 = (a, b | a³ = b² = e, ba = a*b). Let H ((b, 1)) and K = D10 × {0}. (a) Draw a partial lattice diagram of subgroups of G that includes the subgroups H and K. Include also H N 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) What is NG(H), the normalizer of H in G? Why? Is NG(K) = G?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let G = D10 x Z/7Z, where, as usual, D10 = (a, b | a³ = b² = e, ba = a*b). Let H = ((b, 1)) and K = D10 × {0}. (a) Draw a partial lattice diagram of subgroups of G that includes the subgroups H and K. Include also HN 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) What is NG(H), the normalizer of H in G? Why? (c) Is NG(K) = G? (d) What is |clG(b, 1)|? Note that clg(x) is the conjugacy class of x in G and that we are only interested in the order of this set. (e) Can you add the subgroup L = ((a, 0)) to your lattice diagram?
Transcribed Image Text:Let G = D10 x Z/7Z, where, as usual, D10 = (a, b | a³ = b² = e, ba = a*b). Let H = ((b, 1)) and K = D10 × {0}. (a) Draw a partial lattice diagram of subgroups of G that includes the subgroups H and K. Include also HN 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) What is NG(H), the normalizer of H in G? Why? (c) Is NG(K) = G? (d) What is |clG(b, 1)|? Note that clg(x) is the conjugacy class of x in G and that we are only interested in the order of this set. (e) Can you add the subgroup L = ((a, 0)) to your lattice diagram?
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